Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , +) . Neutral element: 0 : = ( 1 , 1 ) , inverse element of � � � � ( a , b ) ( b , a ) : . 2. ( Z m , +) . Neutral element: 0 : = [ 0 ] m logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , +) . Neutral element: 0 : = ( 1 , 1 ) , inverse element of � � � � ( a , b ) ( b , a ) : . 2. ( Z m , +) . Neutral element: 0 : = [ 0 ] m , inverse element of [ a ] m logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , +) . Neutral element: 0 : = ( 1 , 1 ) , inverse element of � � � � ( a , b ) ( b , a ) : . 2. ( Z m , +) . Neutral element: 0 : = [ 0 ] m , inverse element of [ a ] m : [ m − a ] m . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , +) . Neutral element: 0 : = ( 1 , 1 ) , inverse element of � � � � ( a , b ) ( b , a ) : . 2. ( Z m , +) . Neutral element: 0 : = [ 0 ] m , inverse element of [ a ] m : [ m − a ] m . � � Bij ( A ) , ◦ : The group of bijective functions f : A → A . 3. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , +) . Neutral element: 0 : = ( 1 , 1 ) , inverse element of � � � � ( a , b ) ( b , a ) : . 2. ( Z m , +) . Neutral element: 0 : = [ 0 ] m , inverse element of [ a ] m : [ m − a ] m . � � Bij ( A ) , ◦ : The group of bijective functions f : A → A . 3. Neutral element: id A logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , +) . Neutral element: 0 : = ( 1 , 1 ) , inverse element of � � � � ( a , b ) ( b , a ) : . 2. ( Z m , +) . Neutral element: 0 : = [ 0 ] m , inverse element of [ a ] m : [ m − a ] m . � � Bij ( A ) , ◦ : The group of bijective functions f : A → A . 3. Neutral element: id A , inverse element of f logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , +) . Neutral element: 0 : = ( 1 , 1 ) , inverse element of � � � � ( a , b ) ( b , a ) : . 2. ( Z m , +) . Neutral element: 0 : = [ 0 ] m , inverse element of [ a ] m : [ m − a ] m . � � Bij ( A ) , ◦ : The group of bijective functions f : A → A . 3. Neutral element: id A , inverse element of f : f − 1 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , +) . Neutral element: 0 : = ( 1 , 1 ) , inverse element of � � � � ( a , b ) ( b , a ) : . 2. ( Z m , +) . Neutral element: 0 : = [ 0 ] m , inverse element of [ a ] m : [ m − a ] m . � � Bij ( A ) , ◦ : The group of bijective functions f : A → A . 3. Neutral element: id A , inverse element of f : f − 1 . 4. Not every semigroup is a group logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , +) . Neutral element: 0 : = ( 1 , 1 ) , inverse element of � � � � ( a , b ) ( b , a ) : . 2. ( Z m , +) . Neutral element: 0 : = [ 0 ] m , inverse element of [ a ] m : [ m − a ] m . � � Bij ( A ) , ◦ : The group of bijective functions f : A → A . 3. Neutral element: id A , inverse element of f : f − 1 . 4. Not every semigroup is a group: ( N , +) is not a group. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results What Do We Already Know About Commutative Groups In General, And About Z In Particular? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. n n n ∑ ∑ ∑ ( a j + b j ) = a j + 4. Sums can be added: b j . j = 1 j = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. n n n ∑ ∑ ∑ ( a j + b j ) = a j + 4. Sums can be added: b j . j = 1 j = 1 j = 1 n n ∑ ∑ a j = 5. Sums can be reordered: a σ ( j ) . j = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. n n n ∑ ∑ ∑ ( a j + b j ) = a j + 4. Sums can be added: b j . j = 1 j = 1 j = 1 n n ∑ ∑ a j = 5. Sums can be reordered: a σ ( j ) . j = 1 j = 1 n k + n ∑ ∑ a j + k = 6. Sums can be re-indexed: a i . j = 1 i = k + 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. n n n ∑ ∑ ∑ ( a j + b j ) = a j + 4. Sums can be added: b j . j = 1 j = 1 j = 1 n n ∑ ∑ a j = 5. Sums can be reordered: a σ ( j ) . j = 1 j = 1 n k + n ∑ ∑ a j + k = 6. Sums can be re-indexed: a i . j = 1 i = k + 1 n m m ∑ ∑ ∑ 7. Sums can be combined: a j + a j = a j . j = 1 j = n + 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 ◦ . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 . a , a ∈ S satisfy ˜ a ◦ a = a ◦ ˜ a = e and a ◦ a = a ◦ a = e Proof. If ˜ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 . a , a ∈ S satisfy ˜ a ◦ a = a ◦ ˜ a = e and a ◦ a = a ◦ a = e , Proof. If ˜ then logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 . a , a ∈ S satisfy ˜ a ◦ a = a ◦ ˜ a = e and a ◦ a = a ◦ a = e , Proof. If ˜ then a ˜ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 . a , a ∈ S satisfy ˜ a ◦ a = a ◦ ˜ a = e and a ◦ a = a ◦ a = e , Proof. If ˜ then a = ˜ a ◦ e ˜ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 . a , a ∈ S satisfy ˜ a ◦ a = a ◦ ˜ a = e and a ◦ a = a ◦ a = e , Proof. If ˜ then a = ˜ a ◦ e = ˜ a ◦ ( a ◦ a ) ˜ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 . a , a ∈ S satisfy ˜ a ◦ a = a ◦ ˜ a = e and a ◦ a = a ◦ a = e , Proof. If ˜ then a = ˜ a ◦ e = ˜ a ◦ ( a ◦ a ) = ( ˜ a ◦ a ) ◦ a ˜ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 . a , a ∈ S satisfy ˜ a ◦ a = a ◦ ˜ a = e and a ◦ a = a ◦ a = e , Proof. If ˜ then a = ˜ a ◦ e = ˜ a ◦ ( a ◦ a ) = ( ˜ a ◦ a ) ◦ a = e ◦ a ˜ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 . a , a ∈ S satisfy ˜ a ◦ a = a ◦ ˜ a = e and a ◦ a = a ◦ a = e , Proof. If ˜ then a = ˜ a ◦ e = ˜ a ◦ ( a ◦ a ) = ( ˜ a ◦ a ) ◦ a = e ◦ a = a . ˜ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 . a , a ∈ S satisfy ˜ a ◦ a = a ◦ ˜ a = e and a ◦ a = a ◦ a = e , Proof. If ˜ then a = ˜ a ◦ e = ˜ a ◦ ( a ◦ a ) = ( ˜ a ◦ a ) ◦ a = e ◦ a = a . ˜ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations on the set R . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations on the set R . The triple ( R , + , · ) is called a ring iff logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations on the set R . The triple ( R , + , · ) is called a ring iff 1. Addition is associative logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations 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 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations 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 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations 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 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations 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 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations 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 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations 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 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations 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 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations 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 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations 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 over addition logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations 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 over addition, that is, for all α , x , y ∈ R we have logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations 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 over addition, that is, for all α , x , y ∈ R we have α · ( x + y ) = α · x + α · y logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Definition. Let + : R × R → R and · : R × R → R be binary operations 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 over 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 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Moreover, we introduce the following special properties for rings. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results What Do We Already Know About Rings In General, And About Z In Particular? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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(!) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , + , · ) is a commutative ring with unity ( 2 , 1 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , + , · ) is a commutative ring with unity ( 2 , 1 ) . 2. ( Z m , + , · ) is a commutative ring with unity [ 1 ] m . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , + , · ) is a commutative ring with unity ( 2 , 1 ) . 2. ( Z m , + , · ) is a commutative ring with unity [ 1 ] m . (From our visualizations, these things literally are “rings”.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , + , · ) is a commutative ring with unity ( 2 , 1 ) . 2. ( Z m , + , · ) is a commutative ring with unity [ 1 ] m . (From our visualizations, these things literally are “rings”.) 3. Let ( R , + , · ) be a commutative ring. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , + , · ) is a commutative ring with unity ( 2 , 1 ) . 2. ( Z m , + , · ) 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 n ∑ a j x j , where n ∈ N and all a j ∈ R , is of the form f ( x ) = j = 0 called a polynomial . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Groups “Avalanche of Knowledge” Rings New Results Examples � � 1. ( Z , + , · ) is a commutative ring with unity ( 2 , 1 ) . 2. ( Z m , + , · ) 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 n ∑ a j x j , where n ∈ N and all a j ∈ R , is of the form f ( x ) = j = 0 called a polynomial . Let R [ x ] be the set of all polynomials. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Groups and Rings
Recommend
More recommend
