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logo1 Groups “Avalanche of Knowledge” Rings New Results

Groups and Rings

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Definition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let G be a set and let ◦ : G×G → G be a binary
• peration on G.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let G be a set and let ◦ : G×G → G be a binary
• peration on G. Then (G,◦) is called a group iff the following

hold.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let G be a set and let ◦ : G×G → G be a binary
• peration on G. Then (G,◦) is called a group iff the following

hold.

• 1. The operation ◦ is associative.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let G be a set and let ◦ : G×G → G be a binary
• peration on G. Then (G,◦) is called a group iff the following

hold.

• 1. The operation ◦ is associative. That is, for all a,b,c ∈ G

we have that (a◦b)◦c = a◦(b◦c).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let G be a set and let ◦ : G×G → G be a binary
• peration on G. Then (G,◦) is called a group iff the following

hold.

• 1. The operation ◦ is associative. That is, for all a,b,c ∈ G

we have that (a◦b)◦c = a◦(b◦c).

• 2. There is a neutral element e ∈ G with respect to ◦.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let G be a set and let ◦ : G×G → G be a binary
• peration on G. Then (G,◦) is called a group iff the following

hold.

• 1. The operation ◦ is associative. That is, for all a,b,c ∈ G

we have that (a◦b)◦c = a◦(b◦c).

• 2. There is a neutral element e ∈ G with respect to ◦. That

is, there is an e ∈ G so that for all a ∈ G we have that e◦a = a◦e = a.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let G be a set and let ◦ : G×G → G be a binary
• peration on G. Then (G,◦) is called a group iff the following

hold.

• 1. The operation ◦ is associative. That is, for all a,b,c ∈ G

we have that (a◦b)◦c = a◦(b◦c).

• 2. There is a neutral element e ∈ G with respect to ◦. That

is, there is an e ∈ G so that for all a ∈ G we have that e◦a = a◦e = a.

• 3. For every a ∈ G there is an inverse element ˜

a.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let G be a set and let ◦ : G×G → G be a binary
• peration on G. Then (G,◦) is called a group iff the following

hold.

• 1. The operation ◦ is associative. That is, for all a,b,c ∈ G

we have that (a◦b)◦c = a◦(b◦c).

• 2. There is a neutral element e ∈ G with respect to ◦. That

is, there is an e ∈ G so that for all a ∈ G we have that e◦a = a◦e = a.

• 3. For every a ∈ G there is an inverse element ˜
• a. That is, for

every a ∈ G there is an ˜ a ∈ G so that a◦ ˜ a = ˜ a◦a = e.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let G be a set and let ◦ : G×G → G be a binary
• peration on G. Then (G,◦) is called a group iff the following

hold.

• 1. The operation ◦ is associative. That is, for all a,b,c ∈ G

we have that (a◦b)◦c = a◦(b◦c).

• 2. There is a neutral element e ∈ G with respect to ◦. That

is, there is an e ∈ G so that for all a ∈ G we have that e◦a = a◦e = a.

• 3. For every a ∈ G there is an inverse element ˜
• a. That is, for

every a ∈ G there is an ˜ a ∈ G so that a◦ ˜ a = ˜ a◦a = e. Moreover, a group is called commutative iff ◦ is commutative.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let G be a set and let ◦ : G×G → G be a binary
• peration on G. Then (G,◦) is called a group iff the following

hold.

• 1. The operation ◦ is associative. That is, for all a,b,c ∈ G

we have that (a◦b)◦c = a◦(b◦c).

• 2. There is a neutral element e ∈ G with respect to ◦. That

is, there is an e ∈ G so that for all a ∈ G we have that e◦a = a◦e = a.

• 3. For every a ∈ G there is an inverse element ˜
• a. That is, for

every a ∈ G there is an ˜ a ∈ G so that a◦ ˜ a = ˜ a◦a = e. Moreover, a group is called commutative iff ◦ is commutative. That is, iff for all a,b ∈ G we have a◦b = b◦a.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Notation

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Notation

• 1. General groups:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Notation

• 1. General groups: group operation ◦

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Notation

• 1. General groups: group operation ◦, neutral element e

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Notation

• 1. General groups: group operation ◦, neutral element e,

inverse element a−1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Notation

• 1. General groups: group operation ◦, neutral element e,

inverse element a−1.

• 2. Commutative groups:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Notation

• 1. General groups: group operation ◦, neutral element e,

inverse element a−1.

• 2. Commutative groups: group operation +

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Notation

• 1. General groups: group operation ◦, neutral element e,

inverse element a−1.

• 2. Commutative groups: group operation +, neutral element

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Notation

• 1. General groups: group operation ◦, neutral element e,

inverse element a−1.

• 2. Commutative groups: group operation +, neutral element

0, inverse element −a.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+). Neutral element: 0 :=
• (1,1)
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+). Neutral element: 0 :=
• (1,1)
• , inverse element of
• (a,b)
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+). Neutral element: 0 :=
• (1,1)
• , inverse element of
• (a,b)
• :
• (b,a)
• .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+). Neutral element: 0 :=
• (1,1)
• , inverse element of
• (a,b)
• :
• (b,a)
• .
• 2. (Zm,+).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+). Neutral element: 0 :=
• (1,1)
• , inverse element of
• (a,b)
• :
• (b,a)
• .
• 2. (Zm,+). Neutral element: 0 := [0]m

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+). Neutral element: 0 :=
• (1,1)
• , inverse element of
• (a,b)
• :
• (b,a)
• .
• 2. (Zm,+). Neutral element: 0 := [0]m, inverse element of

[a]m

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+). Neutral element: 0 :=
• (1,1)
• , inverse element of
• (a,b)
• :
• (b,a)
• .
• 2. (Zm,+). Neutral element: 0 := [0]m, inverse element of

[a]m: [m−a]m.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+). Neutral element: 0 :=
• (1,1)
• , inverse element of
• (a,b)
• :
• (b,a)
• .
• 2. (Zm,+). Neutral element: 0 := [0]m, inverse element of

[a]m: [m−a]m. 3.

• Bij(A),◦
• : The group of bijective functions f : A → A.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+). Neutral element: 0 :=
• (1,1)
• , inverse element of
• (a,b)
• :
• (b,a)
• .
• 2. (Zm,+). Neutral element: 0 := [0]m, inverse element of

[a]m: [m−a]m. 3.

• Bij(A),◦
• : The group of bijective functions f : A → A.

Neutral element: idA

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+). Neutral element: 0 :=
• (1,1)
• , inverse element of
• (a,b)
• :
• (b,a)
• .
• 2. (Zm,+). Neutral element: 0 := [0]m, inverse element of

[a]m: [m−a]m. 3.

• Bij(A),◦
• : The group of bijective functions f : A → A.

Neutral element: idA, inverse element of f

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+). Neutral element: 0 :=
• (1,1)
• , inverse element of
• (a,b)
• :
• (b,a)
• .
• 2. (Zm,+). Neutral element: 0 := [0]m, inverse element of

[a]m: [m−a]m. 3.

• Bij(A),◦
• : The group of bijective functions f : A → A.

Neutral element: idA, inverse element of f: f −1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+). Neutral element: 0 :=
• (1,1)
• , inverse element of
• (a,b)
• :
• (b,a)
• .
• 2. (Zm,+). Neutral element: 0 := [0]m, inverse element of

[a]m: [m−a]m. 3.

• Bij(A),◦
• : The group of bijective functions f : A → A.

Neutral element: idA, inverse element of f: f −1.

• 4. Not every semigroup is a group

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+). Neutral element: 0 :=
• (1,1)
• , inverse element of
• (a,b)
• :
• (b,a)
• .
• 2. (Zm,+). Neutral element: 0 := [0]m, inverse element of

[a]m: [m−a]m. 3.

• Bij(A),◦
• : The group of bijective functions f : A → A.

Neutral element: idA, inverse element of f: f −1.

• 4. Not every semigroup is a group: (N,+) is not a group.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Commutative Groups In General, And About Z In Particular?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Commutative Groups In General, And About Z In Particular?

• 1. The neutral element (e or 0) is unique.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Commutative Groups In General, And About Z In Particular?

• 1. The neutral element (e or 0) is unique.
• 2. Sums can be defined.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Commutative Groups In General, And About Z In Particular?

• 1. The neutral element (e or 0) is unique.
• 2. Sums can be defined.
• 3. No need to bracket two summands for each addition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Commutative Groups In General, And About Z In Particular?

• 1. The neutral element (e or 0) is unique.
• 2. Sums can be defined.
• 3. No need to bracket two summands for each addition.
• 4. Sums can be added:

n

∑

j=1

(aj +bj) =

n

∑

j=1

aj +

n

∑

j=1

bj.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Commutative Groups In General, And About Z In Particular?

• 1. The neutral element (e or 0) is unique.
• 2. Sums can be defined.
• 3. No need to bracket two summands for each addition.
• 4. Sums can be added:

n

∑

j=1

(aj +bj) =

n

∑

j=1

aj +

n

∑

j=1

bj.

• 5. Sums can be reordered:

n

∑

j=1

aj =

n

∑

j=1

aσ(j).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Commutative Groups In General, And About Z In Particular?

• 1. The neutral element (e or 0) is unique.
• 2. Sums can be defined.
• 3. No need to bracket two summands for each addition.
• 4. Sums can be added:

n

∑

j=1

(aj +bj) =

n

∑

j=1

aj +

n

∑

j=1

bj.

• 5. Sums can be reordered:

n

∑

j=1

aj =

n

∑

j=1

aσ(j).

• 6. Sums can be re-indexed:

n

∑

j=1

aj+k =

k+n

∑

i=k+1

ai.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Commutative Groups In General, And About Z In Particular?

• 1. The neutral element (e or 0) is unique.
• 2. Sums can be defined.
• 3. No need to bracket two summands for each addition.
• 4. Sums can be added:

n

∑

j=1

(aj +bj) =

n

∑

j=1

aj +

n

∑

j=1

bj.

• 5. Sums can be reordered:

n

∑

j=1

aj =

n

∑

j=1

aσ(j).

• 6. Sums can be re-indexed:

n

∑

j=1

aj+k =

k+n

∑

i=k+1

ai.

• 7. Sums can be combined:

n

∑

j=1

aj +

m

∑

j=n+1

aj =

m

∑

j=1

aj.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Theorem. Let (S,◦) be a semigroup with neutral element e and

let a ∈ S have an inverse element with respect to ◦.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Theorem. Let (S,◦) be a semigroup with neutral element e and

let a ∈ S have an inverse element with respect to ◦. Then a has exactly one inverse element.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Theorem. Let (S,◦) be a semigroup with neutral element e and

let a ∈ S have an inverse element with respect to ◦. Then a has exactly one inverse element. That is, if ˜ a and a both have the properties of an inverse element of a, then ˜ a = a.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Theorem. Let (S,◦) be a semigroup with neutral element e and

let a ∈ S have an inverse element with respect to ◦. Then a has exactly one inverse element. That is, if ˜ a and a both have the properties of an inverse element of a, then ˜ a = a. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Theorem. Let (S,◦) be a semigroup with neutral element e and

let a ∈ S have an inverse element with respect to ◦. Then a has exactly one inverse element. That is, if ˜ a and a both have the properties of an inverse element of a, then ˜ a = a.

• Proof. If ˜

a,a ∈ S satisfy ˜ a◦a = a◦ ˜ a = e and a◦a = a◦a = e

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Theorem. Let (S,◦) be a semigroup with neutral element e and

let a ∈ S have an inverse element with respect to ◦. Then a has exactly one inverse element. That is, if ˜ a and a both have the properties of an inverse element of a, then ˜ a = a.

• Proof. If ˜

a,a ∈ S satisfy ˜ a◦a = a◦ ˜ a = e and a◦a = a◦a = e, then

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Theorem. Let (S,◦) be a semigroup with neutral element e and

let a ∈ S have an inverse element with respect to ◦. Then a has exactly one inverse element. That is, if ˜ a and a both have the properties of an inverse element of a, then ˜ a = a.

• Proof. If ˜

a,a ∈ S satisfy ˜ a◦a = a◦ ˜ a = e and a◦a = a◦a = e, then ˜ a

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Theorem. Let (S,◦) be a semigroup with neutral element e and

let a ∈ S have an inverse element with respect to ◦. Then a has exactly one inverse element. That is, if ˜ a and a both have the properties of an inverse element of a, then ˜ a = a.

• Proof. If ˜

a,a ∈ S satisfy ˜ a◦a = a◦ ˜ a = e and a◦a = a◦a = e, then ˜ a = ˜ a◦e

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Theorem. Let (S,◦) be a semigroup with neutral element e and

let a ∈ S have an inverse element with respect to ◦. Then a has exactly one inverse element. That is, if ˜ a and a both have the properties of an inverse element of a, then ˜ a = a.

• Proof. If ˜

a,a ∈ S satisfy ˜ a◦a = a◦ ˜ a = e and a◦a = a◦a = e, then ˜ a = ˜ a◦e = ˜ a◦(a◦a)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Theorem. Let (S,◦) be a semigroup with neutral element e and

let a ∈ S have an inverse element with respect to ◦. Then a has exactly one inverse element. That is, if ˜ a and a both have the properties of an inverse element of a, then ˜ a = a.

• Proof. If ˜

a,a ∈ S satisfy ˜ a◦a = a◦ ˜ a = e and a◦a = a◦a = e, then ˜ a = ˜ a◦e = ˜ a◦(a◦a) = (˜ a◦a)◦a

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Theorem. Let (S,◦) be a semigroup with neutral element e and

let a ∈ S have an inverse element with respect to ◦. Then a has exactly one inverse element. That is, if ˜ a and a both have the properties of an inverse element of a, then ˜ a = a.

• Proof. If ˜

a,a ∈ S satisfy ˜ a◦a = a◦ ˜ a = e and a◦a = a◦a = e, then ˜ a = ˜ a◦e = ˜ a◦(a◦a) = (˜ a◦a)◦a = e◦a

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Theorem. Let (S,◦) be a semigroup with neutral element e and

let a ∈ S have an inverse element with respect to ◦. Then a has exactly one inverse element. That is, if ˜ a and a both have the properties of an inverse element of a, then ˜ a = a.

• Proof. If ˜

a,a ∈ S satisfy ˜ a◦a = a◦ ˜ a = e and a◦a = a◦a = e, then ˜ a = ˜ a◦e = ˜ a◦(a◦a) = (˜ a◦a)◦a = e◦a = a.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Theorem. Let (S,◦) be a semigroup with neutral element e and

let a ∈ S have an inverse element with respect to ◦. Then a has exactly one inverse element. That is, if ˜ a and a both have the properties of an inverse element of a, then ˜ a = a.

• Proof. If ˜

a,a ∈ S satisfy ˜ a◦a = a◦ ˜ a = e and a◦a = a◦a = e, then ˜ a = ˜ a◦e = ˜ a◦(a◦a) = (˜ a◦a)◦a = e◦a = a.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Definition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff
• 1. Addition is associative

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff
• 1. Addition is associative, that is, for all x,y,z ∈ R we have

(x+y)+z = x+(y+z).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff
• 1. Addition is associative, that is, for all x,y,z ∈ R we have

(x+y)+z = x+(y+z).

• 2. Addition is commutative

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff
• 1. Addition is associative, that is, for all x,y,z ∈ R we have

(x+y)+z = x+(y+z).

• 2. Addition is commutative, that is, for all x,y ∈ R we have

x+y = y+x.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff
• 1. Addition is associative, that is, for all x,y,z ∈ R we have

(x+y)+z = x+(y+z).

• 2. Addition is commutative, that is, for all x,y ∈ R we have

x+y = y+x.

• 3. There is a neutral element 0 for addition

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff
• 1. Addition is associative, that is, for all x,y,z ∈ R we have

(x+y)+z = x+(y+z).

• 2. Addition is commutative, that is, for all x,y ∈ R we have

x+y = y+x.

• 3. There is a neutral element 0 for addition, that is, there is

an element 0 ∈ R so that for all x ∈ R we have x+0 = x.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff
• 1. Addition is associative, that is, for all x,y,z ∈ R we have

(x+y)+z = x+(y+z).

• 2. Addition is commutative, that is, for all x,y ∈ R we have

x+y = y+x.

• 3. There is a neutral element 0 for addition, that is, there is

an element 0 ∈ R so that for all x ∈ R we have x+0 = x.

• 4. For every element x ∈ R there is an additive inverse

element (−x) so that x+(−x) = 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff
• 1. Addition is associative, that is, for all x,y,z ∈ R we have

(x+y)+z = x+(y+z).

• 2. Addition is commutative, that is, for all x,y ∈ R we have

x+y = y+x.

• 3. There is a neutral element 0 for addition, that is, there is

an element 0 ∈ R so that for all x ∈ R we have x+0 = x.

• 4. For every element x ∈ R there is an additive inverse

element (−x) so that x+(−x) = 0.

• 5. Multiplication is associative

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff
• 1. Addition is associative, that is, for all x,y,z ∈ R we have

(x+y)+z = x+(y+z).

• 2. Addition is commutative, that is, for all x,y ∈ R we have

x+y = y+x.

• 3. There is a neutral element 0 for addition, that is, there is

an element 0 ∈ R so that for all x ∈ R we have x+0 = x.

• 4. For every element x ∈ R there is an additive inverse

element (−x) so that x+(−x) = 0.

• 5. Multiplication is associative, that is, for all x,y,z ∈ R we

have (x·y)·z = x·(y·z).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff
• 1. Addition is associative, that is, for all x,y,z ∈ R we have

(x+y)+z = x+(y+z).

• 2. Addition is commutative, that is, for all x,y ∈ R we have

x+y = y+x.

• 3. There is a neutral element 0 for addition, that is, there is

an element 0 ∈ R so that for all x ∈ R we have x+0 = x.

• 4. For every element x ∈ R there is an additive inverse

element (−x) so that x+(−x) = 0.

• 5. Multiplication is associative, that is, for all x,y,z ∈ R we

have (x·y)·z = x·(y·z).

• 6. Multiplication is left distributive and right distributive
• ver addition

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff
• 1. Addition is associative, that is, for all x,y,z ∈ R we have

(x+y)+z = x+(y+z).

• 2. Addition is commutative, that is, for all x,y ∈ R we have

x+y = y+x.

• 3. There is a neutral element 0 for addition, that is, there is

an element 0 ∈ R so that for all x ∈ R we have x+0 = x.

• 4. For every element x ∈ R there is an additive inverse

element (−x) so that x+(−x) = 0.

• 5. Multiplication is associative, that is, for all x,y,z ∈ R we

have (x·y)·z = x·(y·z).

• 6. Multiplication is left distributive and right distributive
• ver addition, that is, for all α,x,y ∈ R we have

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff
• 1. Addition is associative, that is, for all x,y,z ∈ R we have

(x+y)+z = x+(y+z).

• 2. Addition is commutative, that is, for all x,y ∈ R we have

x+y = y+x.

• 3. There is a neutral element 0 for addition, that is, there is

an element 0 ∈ R so that for all x ∈ R we have x+0 = x.

• 4. For every element x ∈ R there is an additive inverse

element (−x) so that x+(−x) = 0.

• 5. Multiplication is associative, that is, for all x,y,z ∈ R we

have (x·y)·z = x·(y·z).

• 6. Multiplication is left distributive and right distributive
• ver addition, that is, for all α,x,y ∈ R we have

α ·(x+y) = α ·x+α ·y

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Definition. Let + : R×R → R and · : R×R → R be binary
• perations on the set R. The triple (R,+,·) is called a ring iff
• 1. Addition is associative, that is, for all x,y,z ∈ R we have

(x+y)+z = x+(y+z).

• 2. Addition is commutative, that is, for all x,y ∈ R we have

x+y = y+x.

• 3. There is a neutral element 0 for addition, that is, there is

an element 0 ∈ R so that for all x ∈ R we have x+0 = x.

• 4. For every element x ∈ R there is an additive inverse

element (−x) so that x+(−x) = 0.

• 5. Multiplication is associative, that is, for all x,y,z ∈ R we

have (x·y)·z = x·(y·z).

• 6. Multiplication is left distributive and right distributive
• ver addition, that is, for all α,x,y ∈ R we have

α ·(x+y) = α ·x+α ·y and (x+y)·α = x·α +y·α.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Moreover, we introduce the following special properties for rings.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Moreover, we introduce the following special properties for rings.

• 7. A ring is called commutative iff multiplication is

commutative

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Moreover, we introduce the following special properties for rings.

• 7. A ring is called commutative iff multiplication is

commutative, that is, iff for all x,y ∈ R we have x·y = y·x.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Moreover, we introduce the following special properties for rings.

• 7. A ring is called commutative iff multiplication is

commutative, that is, iff for all x,y ∈ R we have x·y = y·x.

• 8. A ring is called a ring with unity iff there is a neutral

element 1 = 0 for multiplication

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Moreover, we introduce the following special properties for rings.

• 7. A ring is called commutative iff multiplication is

commutative, that is, iff for all x,y ∈ R we have x·y = y·x.

• 8. A ring is called a ring with unity iff there is a neutral

element 1 = 0 for multiplication, that is, iff there is an element 1 ∈ R\{0} so that for all x ∈ R we have 1·x = x.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Moreover, we introduce the following special properties for rings.

• 7. A ring is called commutative iff multiplication is

commutative, that is, iff for all x,y ∈ R we have x·y = y·x.

• 8. A ring is called a ring with unity iff there is a neutral

element 1 = 0 for multiplication, that is, iff there is an element 1 ∈ R\{0} so that for all x ∈ R we have 1·x = x.

• 9. In a ring with unity, an element b is called a multiplicative

inverse of the element a iff ab = ba = 1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Rings In General, And About Z In Particular?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Rings In General, And About Z In Particular?

• 1. Everything we knew about commutative groups.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Rings In General, And About Z In Particular?

• 1. Everything we knew about commutative groups. (Unique

neutral and inverse elements, properties of summations.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Rings In General, And About Z In Particular?

• 1. Everything we knew about commutative groups. (Unique

neutral and inverse elements, properties of summations.)

• 2. No need for parentheses in long products.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Rings In General, And About Z In Particular?

• 1. Everything we knew about commutative groups. (Unique

neutral and inverse elements, properties of summations.)

• 2. No need for parentheses in long products.
• 3. Parentheses are multiplied out as usual.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Rings In General, And About Z In Particular?

• 1. Everything we knew about commutative groups. (Unique

neutral and inverse elements, properties of summations.)

• 2. No need for parentheses in long products.
• 3. Parentheses are multiplied out as usual.
• 4. Can define products and powers.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Rings In General, And About Z In Particular?

• 1. Everything we knew about commutative groups. (Unique

neutral and inverse elements, properties of summations.)

• 2. No need for parentheses in long products.
• 3. Parentheses are multiplied out as usual.
• 4. Can define products and powers.
• 5. In commutative rings:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Rings In General, And About Z In Particular?

• 1. Everything we knew about commutative groups. (Unique

neutral and inverse elements, properties of summations.)

• 2. No need for parentheses in long products.
• 3. Parentheses are multiplied out as usual.
• 4. Can define products and powers.
• 5. In commutative rings: Power laws

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Rings In General, And About Z In Particular?

• 1. Everything we knew about commutative groups. (Unique

neutral and inverse elements, properties of summations.)

• 2. No need for parentheses in long products.
• 3. Parentheses are multiplied out as usual.
• 4. Can define products and powers.
• 5. In commutative rings: Power laws and, if it is a ring with

unity, the Binomial Theorem

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Rings In General, And About Z In Particular?

• 1. Everything we knew about commutative groups. (Unique

neutral and inverse elements, properties of summations.)

• 2. No need for parentheses in long products.
• 3. Parentheses are multiplied out as usual.
• 4. Can define products and powers.
• 5. In commutative rings: Power laws and, if it is a ring with

unity, the Binomial Theorem(!)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Rings In General, And About Z In Particular?

• 1. Everything we knew about commutative groups. (Unique

neutral and inverse elements, properties of summations.)

• 2. No need for parentheses in long products.
• 3. Parentheses are multiplied out as usual.
• 4. Can define products and powers.
• 5. In commutative rings: Power laws and, if it is a ring with

unity, the Binomial Theorem(!)

• 6. In rings with unity:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Rings In General, And About Z In Particular?

• 1. Everything we knew about commutative groups. (Unique

neutral and inverse elements, properties of summations.)

• 2. No need for parentheses in long products.
• 3. Parentheses are multiplied out as usual.
• 4. Can define products and powers.
• 5. In commutative rings: Power laws and, if it is a ring with

unity, the Binomial Theorem(!)

• 6. In rings with unity: The neutral element 1 is unique

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

What Do We Already Know About Rings In General, And About Z In Particular?

• 1. Everything we knew about commutative groups. (Unique

neutral and inverse elements, properties of summations.)

• 2. No need for parentheses in long products.
• 3. Parentheses are multiplied out as usual.
• 4. Can define products and powers.
• 5. In commutative rings: Power laws and, if it is a ring with

unity, the Binomial Theorem(!)

• 6. In rings with unity: The neutral element 1 is unique, any

existing multiplicative inverses are unique.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+,·) is a commutative ring with unity
• (2,1)
• .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+,·) is a commutative ring with unity
• (2,1)
• .
• 2. (Zm,+,·) is a commutative ring with unity [1]m.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+,·) is a commutative ring with unity
• (2,1)
• .
• 2. (Zm,+,·) is a commutative ring with unity [1]m.

(From our visualizations, these things literally are “rings”.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+,·) is a commutative ring with unity
• (2,1)
• .
• 2. (Zm,+,·) is a commutative ring with unity [1]m.

(From our visualizations, these things literally are “rings”.)

• 3. Let (R,+,·) be a commutative ring.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+,·) is a commutative ring with unity
• (2,1)
• .
• 2. (Zm,+,·) is a commutative ring with unity [1]m.

(From our visualizations, these things literally are “rings”.)

• 3. Let (R,+,·) be a commutative ring. A function f : R → R
• f the form f(x) =

n

∑

j=0

ajxj, where n ∈ N and all aj ∈ R, is called a polynomial.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+,·) is a commutative ring with unity
• (2,1)
• .
• 2. (Zm,+,·) is a commutative ring with unity [1]m.

(From our visualizations, these things literally are “rings”.)

• 3. Let (R,+,·) be a commutative ring. A function f : R → R
• f the form f(x) =

n

∑

j=0

ajxj, where n ∈ N and all aj ∈ R, is called a polynomial. Let R[x] be the set of all polynomials.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+,·) is a commutative ring with unity
• (2,1)
• .
• 2. (Zm,+,·) is a commutative ring with unity [1]m.

(From our visualizations, these things literally are “rings”.)

• 3. Let (R,+,·) be a commutative ring. A function f : R → R
• f the form f(x) =

n

∑

j=0

ajxj, where n ∈ N and all aj ∈ R, is called a polynomial. Let R[x] be the set of all polynomials. Let f,g ∈ R[x].

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+,·) is a commutative ring with unity
• (2,1)
• .
• 2. (Zm,+,·) is a commutative ring with unity [1]m.

(From our visualizations, these things literally are “rings”.)

• 3. Let (R,+,·) be a commutative ring. A function f : R → R
• f the form f(x) =

n

∑

j=0

ajxj, where n ∈ N and all aj ∈ R, is called a polynomial. Let R[x] be the set of all polynomials. Let f,g ∈ R[x]. Define addition pointwise as (f +g)(x) := f(x)+g(x)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

• 1. (Z,+,·) is a commutative ring with unity
• (2,1)
• .
• 2. (Zm,+,·) is a commutative ring with unity [1]m.

(From our visualizations, these things literally are “rings”.)

• 3. Let (R,+,·) be a commutative ring. A function f : R → R
• f the form f(x) =

n

∑

j=0

ajxj, where n ∈ N and all aj ∈ R, is called a polynomial. Let R[x] be the set of all polynomials. Let f,g ∈ R[x]. Define addition pointwise as (f +g)(x) := f(x)+g(x) and define multiplication pointwise as (f ·g)(x) :=

• f(x)
• ·
• g(x)
• .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

With f(x) =

n

∑

j=0

ajxj, g(x) =

m

∑

j=0

bjxj,

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

With f(x) =

n

∑

j=0

ajxj, g(x) =

m

∑

j=0

bjxj, M being the larger of m and n,

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

With f(x) =

n

∑

j=0

ajxj, g(x) =

m

∑

j=0

bjxj, M being the larger of m and n, aj := 0 for j ∈ {n+1,...,M} and bj := 0 for j ∈ {m+1,...,M}

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

With f(x) =

n

∑

j=0

ajxj, g(x) =

m

∑

j=0

bjxj, M being the larger of m and n, aj := 0 for j ∈ {n+1,...,M} and bj := 0 for j ∈ {m+1,...,M} we have that (f +g)(x) =

M

∑

j=0

(aj +bj)xj

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

With f(x) =

n

∑

j=0

ajxj, g(x) =

m

∑

j=0

bjxj, M being the larger of m and n, aj := 0 for j ∈ {n+1,...,M} and bj := 0 for j ∈ {m+1,...,M} we have that (f +g)(x) =

M

∑

j=0

(aj +bj)xj and (f ·g)(x) =

2M

∑

j=0

• min{j,M}

∑

k=max{0,j−M}

akbj−k

• xj.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

With f(x) =

n

∑

j=0

ajxj, g(x) =

m

∑

j=0

bjxj, M being the larger of m and n, aj := 0 for j ∈ {n+1,...,M} and bj := 0 for j ∈ {m+1,...,M} we have that (f +g)(x) =

M

∑

j=0

(aj +bj)xj and (f ·g)(x) =

2M

∑

j=0

• min{j,M}

∑

k=max{0,j−M}

akbj−k

• xj. Thus
• R[x],+,·) is a commutative ring

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

With f(x) =

n

∑

j=0

ajxj, g(x) =

m

∑

j=0

bjxj, M being the larger of m and n, aj := 0 for j ∈ {n+1,...,M} and bj := 0 for j ∈ {m+1,...,M} we have that (f +g)(x) =

M

∑

j=0

(aj +bj)xj and (f ·g)(x) =

2M

∑

j=0

• min{j,M}

∑

k=max{0,j−M}

akbj−k

• xj. Thus
• R[x],+,·) is a commutative ring, called the ring of

polynomials over R.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Examples

With f(x) =

n

∑

j=0

ajxj, g(x) =

m

∑

j=0

bjxj, M being the larger of m and n, aj := 0 for j ∈ {n+1,...,M} and bj := 0 for j ∈ {m+1,...,M} we have that (f +g)(x) =

M

∑

j=0

(aj +bj)xj and (f ·g)(x) =

2M

∑

j=0

• min{j,M}

∑

k=max{0,j−M}

akbj−k

• xj. Thus
• R[x],+,·) is a commutative ring, called the ring of

polynomials over R. Moreover, if R is a ring with unity, then so is R[x].

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

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Examples

With f(x) =

n

∑

j=0

ajxj, g(x) =

m

∑

j=0

bjxj, M being the larger of m and n, aj := 0 for j ∈ {n+1,...,M} and bj := 0 for j ∈ {m+1,...,M} we have that (f +g)(x) =

M

∑

j=0

(aj +bj)xj and (f ·g)(x) =

2M

∑

j=0

• min{j,M}

∑

k=max{0,j−M}

akbj−k

• xj. Thus
• R[x],+,·) is a commutative ring, called the ring of

polynomials over R. Moreover, if R is a ring with unity, then so is R[x]. 4.

• Bij(A),◦
• is a commutative group that is not a ring.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Groups and Rings

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Subtraction

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Subtraction

Let (R,+,·) be a ring.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Subtraction

Let (R,+,·) be a ring. We define the binary operation of subtraction for all (x,y) ∈ R×R as x−y := x+(−y).

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Subtraction

Let (R,+,·) be a ring. We define the binary operation of subtraction for all (x,y) ∈ R×R as x−y := x+(−y). The element x−y is also called the difference of x and y.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Groups and Rings

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Integers Modulo m (Customary Definition)

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Groups and Rings

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Integers Modulo m (Customary Definition)

Let m ∈ N.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Groups and Rings

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Integers Modulo m (Customary Definition)

Let m ∈ N.

• 1. x ≡ y (mod m) iff there is a k ∈ Z so that x−y = km.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Integers Modulo m (Customary Definition)

Let m ∈ N.

• 1. x ≡ y (mod m) iff there is a k ∈ Z so that x−y = km.
• 2. Zm is the set of all equivalence classes [x]m of Z with

respect to ≡ (mod m).

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Integers Modulo m (Customary Definition)

Let m ∈ N.

• 1. x ≡ y (mod m) iff there is a k ∈ Z so that x−y = km.
• 2. Zm is the set of all equivalence classes [x]m of Z with

respect to ≡ (mod m).

• 3. Addition [x]m +[y]m := [x+y]m and multiplication

[x]m ·[y]m := [x·y]m are well-defined.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Integers Modulo m (Customary Definition)

Let m ∈ N.

• 1. x ≡ y (mod m) iff there is a k ∈ Z so that x−y = km.
• 2. Zm is the set of all equivalence classes [x]m of Z with

respect to ≡ (mod m).

• 3. Addition [x]m +[y]m := [x+y]m and multiplication

[x]m ·[y]m := [x·y]m are well-defined.

• 4. (Zm,+,·) is a commutative ring with unity.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Integers Modulo m (Customary Definition)

Let m ∈ N.

• 1. x ≡ y (mod m) iff there is a k ∈ Z so that x−y = km.
• 2. Zm is the set of all equivalence classes [x]m of Z with

respect to ≡ (mod m).

• 3. Addition [x]m +[y]m := [x+y]m and multiplication

[x]m ·[y]m := [x·y]m are well-defined.

• 4. (Zm,+,·) is a commutative ring with unity.
• 5. It is “the same” (isomorphic to) the integers modulo m that

we have investigated already.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Integers Modulo m (Customary Definition)

Let m ∈ N.

• 1. x ≡ y (mod m) iff there is a k ∈ Z so that x−y = km.
• 2. Zm is the set of all equivalence classes [x]m of Z with

respect to ≡ (mod m).

• 3. Addition [x]m +[y]m := [x+y]m and multiplication

[x]m ·[y]m := [x·y]m are well-defined.

• 4. (Zm,+,·) is a commutative ring with unity.
• 5. It is “the same” (isomorphic to) the integers modulo m that

we have investigated already. (The new equivalence classes are a bit larger and more convenient, that’s it.)

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Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Proposition.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let a,b,c ∈ R be so that

a+c = b+c.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let a,b,c ∈ R be so that

a+c = b+c. Then a = b.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let a,b,c ∈ R be so that

a+c = b+c. Then a = b. (Not a new result, but a new proof.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let a,b,c ∈ R be so that

a+c = b+c. Then a = b. (Not a new result, but a new proof.) Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let a,b,c ∈ R be so that

a+c = b+c. Then a = b. (Not a new result, but a new proof.) Proof. a

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let a,b,c ∈ R be so that

a+c = b+c. Then a = b. (Not a new result, but a new proof.) Proof. a = a+0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let a,b,c ∈ R be so that

a+c = b+c. Then a = b. (Not a new result, but a new proof.) Proof. a = a+0 = a+(c−c)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let a,b,c ∈ R be so that

a+c = b+c. Then a = b. (Not a new result, but a new proof.) Proof. a = a+0 = a+(c−c) = (a+c)−c

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let a,b,c ∈ R be so that

a+c = b+c. Then a = b. (Not a new result, but a new proof.) Proof. a = a+0 = a+(c−c) = (a+c)−c = (b+c)−c

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let a,b,c ∈ R be so that

a+c = b+c. Then a = b. (Not a new result, but a new proof.) Proof. a = a+0 = a+(c−c) = (a+c)−c = (b+c)−c = b+(c−c)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let a,b,c ∈ R be so that

a+c = b+c. Then a = b. (Not a new result, but a new proof.) Proof. a = a+0 = a+(c−c) = (a+c)−c = (b+c)−c = b+(c−c) = b+0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let a,b,c ∈ R be so that

a+c = b+c. Then a = b. (Not a new result, but a new proof.) Proof. a = a+0 = a+(c−c) = (a+c)−c = (b+c)−c = b+(c−c) = b+0 = b

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let a,b,c ∈ R be so that

a+c = b+c. Then a = b. (Not a new result, but a new proof.) Proof. a = a+0 = a+(c−c) = (a+c)−c = (b+c)−c = b+(c−c) = b+0 = b

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Proposition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R. Then

0·x = x·0 = 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R. Then

0·x = x·0 = 0. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R. Then

0·x = x·0 = 0.

• Proof. Note that 0x+0x

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R. Then

0·x = x·0 = 0.

• Proof. Note that 0x+0x = (0+0)x

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R. Then

0·x = x·0 = 0.

• Proof. Note that 0x+0x = (0+0)x = 0x.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R. Then

0·x = x·0 = 0.

• Proof. Note that 0x+0x = (0+0)x = 0x. Now

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R. Then

0·x = x·0 = 0.

• Proof. Note that 0x+0x = (0+0)x = 0x. Now

= 0x+(−0x)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R. Then

0·x = x·0 = 0.

• Proof. Note that 0x+0x = (0+0)x = 0x. Now

= 0x+(−0x) = (0x+0x)+(−0x)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R. Then

0·x = x·0 = 0.

• Proof. Note that 0x+0x = (0+0)x = 0x. Now

= 0x+(−0x) = (0x+0x)+(−0x) = 0x+

• 0x+(−0x)
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R. Then

0·x = x·0 = 0.

• Proof. Note that 0x+0x = (0+0)x = 0x. Now

= 0x+(−0x) = (0x+0x)+(−0x) = 0x+

• 0x+(−0x)
• = 0x+0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R. Then

0·x = x·0 = 0.

• Proof. Note that 0x+0x = (0+0)x = 0x. Now

= 0x+(−0x) = (0x+0x)+(−0x) = 0x+

• 0x+(−0x)
• = 0x+0 = 0x.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R. Then

0·x = x·0 = 0.

• Proof. Note that 0x+0x = (0+0)x = 0x. Now

= 0x+(−0x) = (0x+0x)+(−0x) = 0x+

• 0x+(−0x)
• = 0x+0 = 0x.

x0 = 0 is proved similarly.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x ∈ R. Then

0·x = x·0 = 0.

• Proof. Note that 0x+0x = (0+0)x = 0x. Now

= 0x+(−0x) = (0x+0x)+(−0x) = 0x+

• 0x+(−0x)
• = 0x+0 = 0x.

x0 = 0 is proved similarly.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

Proposition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

• Proof. By uniqueness of additive inverses, the result is proved

if we can prove that (−x)(−y) is the additive inverse of (−xy).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

• Proof. By uniqueness of additive inverses, the result is proved

if we can prove that (−x)(−y) is the additive inverse of (−xy). Because (−x)y+xy

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

• Proof. By uniqueness of additive inverses, the result is proved

if we can prove that (−x)(−y) is the additive inverse of (−xy). Because (−x)y+xy =

• (−x)+x
• y

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

• Proof. By uniqueness of additive inverses, the result is proved

if we can prove that (−x)(−y) is the additive inverse of (−xy). Because (−x)y+xy =

• (−x)+x
• y = 0y

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

• Proof. By uniqueness of additive inverses, the result is proved

if we can prove that (−x)(−y) is the additive inverse of (−xy). Because (−x)y+xy =

• (−x)+x
• y = 0y = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

• Proof. By uniqueness of additive inverses, the result is proved

if we can prove that (−x)(−y) is the additive inverse of (−xy). Because (−x)y+xy =

• (−x)+x
• y = 0y = 0 we have that

−xy = (−x)y.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

• Proof. By uniqueness of additive inverses, the result is proved

if we can prove that (−x)(−y) is the additive inverse of (−xy). Because (−x)y+xy =

• (−x)+x
• y = 0y = 0 we have that

−xy = (−x)y. But then (−x)(−y)+(−xy)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

• Proof. By uniqueness of additive inverses, the result is proved

if we can prove that (−x)(−y) is the additive inverse of (−xy). Because (−x)y+xy =

• (−x)+x
• y = 0y = 0 we have that

−xy = (−x)y. But then (−x)(−y)+(−xy) = (−x)(−y)+(−x)y

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

• Proof. By uniqueness of additive inverses, the result is proved

if we can prove that (−x)(−y) is the additive inverse of (−xy). Because (−x)y+xy =

• (−x)+x
• y = 0y = 0 we have that

−xy = (−x)y. But then (−x)(−y)+(−xy) = (−x)(−y)+(−x)y = (−x)

• (−y)+y
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

• Proof. By uniqueness of additive inverses, the result is proved

if we can prove that (−x)(−y) is the additive inverse of (−xy). Because (−x)y+xy =

• (−x)+x
• y = 0y = 0 we have that

−xy = (−x)y. But then (−x)(−y)+(−xy) = (−x)(−y)+(−x)y = (−x)

• (−y)+y
• =

(−x)0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

• Proof. By uniqueness of additive inverses, the result is proved

if we can prove that (−x)(−y) is the additive inverse of (−xy). Because (−x)y+xy =

• (−x)+x
• y = 0y = 0 we have that

−xy = (−x)y. But then (−x)(−y)+(−xy) = (−x)(−y)+(−x)y = (−x)

• (−y)+y
• =

(−x)0 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

• Proof. By uniqueness of additive inverses, the result is proved

if we can prove that (−x)(−y) is the additive inverse of (−xy). Because (−x)y+xy =

• (−x)+x
• y = 0y = 0 we have that

−xy = (−x)y. But then (−x)(−y)+(−xy) = (−x)(−y)+(−x)y = (−x)

• (−y)+y
• =

(−x)0 = and we are done.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings

logo1 Groups “Avalanche of Knowledge” Rings New Results

• Proposition. Let (R,+,·) be a ring and let x,y ∈ R. Then

(−x)(−y) = xy.

• Proof. By uniqueness of additive inverses, the result is proved

if we can prove that (−x)(−y) is the additive inverse of (−xy). Because (−x)y+xy =

• (−x)+x
• y = 0y = 0 we have that

−xy = (−x)y. But then (−x)(−y)+(−xy) = (−x)(−y)+(−x)y = (−x)

• (−y)+y
• =

(−x)0 = and we are done.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Groups and Rings