-factorization and -elasticity Richard Hasenauer Bethany Kubik 1 - - PowerPoint PPT Presentation

τ-factorization and τ-elasticity

Richard Hasenauer Bethany Kubik

1Northeastern State University 2University of Minnesota Duluth

22 March 2019

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Definition

Let R be a PID (commutative with identity) and I be an ideal of R. For any nonzero non-unit a P R, we say a “ λb1 ¨ ¨ ¨ bn is a τI-factorization of a if λ is a unit, b1, . . . , bn are nonzero non-units, and b1 ” ¨ ¨ ¨ ” bn pmod Iq.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Example

Let R “ Z and I “ p2q. Then 20 “ 2 ¨ 10 is a τI-factorization since 2 ” 10 pmod 2q.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Example

Let R “ Z and I “ p2q. Then 20 “ 2 ¨ 10 is a τI-factorization since 2 ” 10 pmod 2q. However, 20 “ 4 ¨ 5 is not a τI-factorization since 4 ı 5 pmod 2q.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Example

Let R “ Z and I “ p7q. Then 30 “ 2 ¨ 3 ¨ 5 “ 6 ¨ 5 “ 2 ¨ 15 “ 3 ¨ 10. The only valid τI-factorization of the above list is 3 ¨ 10 since 3 ” 10 pmod 7q.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Definition

We say a P R is a τI-atom if, for any τI-factorization a “ bc, either b or c is a unit.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Definition

We say a P R is a τI-atom if, for any τI-factorization a “ bc, either b or c is a unit.

Definition

We say R is τI-atomic if every nonzero non-unit element has a τI-factorization into a finite product of τI-atoms.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Definition

We say a P R is a τI-atom if, for any τI-factorization a “ bc, either b or c is a unit.

Definition

We say R is τI-atomic if every nonzero non-unit element has a τI-factorization into a finite product of τI-atoms.

Example

Let R “ Z and I “ p1q “ Z. Then R is τI-atomic.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Example

Let R “ Z and I “ p7q. Then 44 “ 4 ¨ 11 is a τI-factorization since 4 ” 11 pmod 7q.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Example

Let R “ Z and I “ p7q. Then 44 “ 4 ¨ 11 is a τI-factorization since 4 ” 11 pmod 7q. Also, 4 “ 2 ¨ 2 is a τI-factorization.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Example

Let R “ Z and I “ p7q. Then 44 “ 4 ¨ 11 is a τI-factorization since 4 ” 11 pmod 7q. Also, 4 “ 2 ¨ 2 is a τI-factorization. However, 44 “ 2 ¨ 2 ¨ 11 is not a τI-factorization.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Example

Let R “ Z and I “ p7q. Then 44 “ 4 ¨ 11 is a τI-factorization since 4 ” 11 pmod 7q. Also, 4 “ 2 ¨ 2 is a τI-factorization. However, 44 “ 2 ¨ 2 ¨ 11 is not a τI-factorization. Since 44 does not factor into a product of τI-atoms, it follows that R is not τI-atomic.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Question: What effect (if any) does the size of R{I have on τI-factorization?

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Question: What effect (if any) does the size of R{I have on τI-factorization?

Fact

If |R{I| “ 2 or 3, then R is always τI-atomic.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Question: What effect (if any) does the size of R{I have on τI-factorization?

Fact

If |R{I| “ 2 or 3, then R is always τI-atomic. The first interesting cases occur when |R{I| “ 4.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Question: What effect (if any) does the size of R{I have on τI-factorization?

Fact

If |R{I| “ 2 or 3, then R is always τI-atomic. The first interesting cases occur when |R{I| “ 4. Commutative rings with identity and four elements: Z4, F4, Z2rxs{px2 ` xq, Z2rxs{px2 ` 1q.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

We assume R is a PID throughout.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

We assume R is a PID throughout.

Fact

R{I is a domain if and only if I is prime.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

We assume R is a PID throughout.

Fact

R{I is a domain if and only if I is prime. In other words, R{I is not a domain if and only if I is not prime.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

We assume R is a PID throughout.

Fact

R{I is a domain if and only if I is prime. In other words, R{I is not a domain if and only if I is not prime.

Remark

When R{I is not a domain, I is not prime. Since R is a PID, we have I “ paq for some non prime a P R. Thus there is no prime p with p ” 0 pmod Iq.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Lemma

Let R be a PID and I an ideal of R. If R{I – Z4, then R is τI-atomic.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Lemma

Let R be a PID and I an ideal of R. If R{I – Z4, then R is τI-atomic. Since R{I – Z4, primes must be equivalent to 1, 2, or 3 pmod Iq.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Lemma

Let R be a PID and I an ideal of R. If R{I – Z4, then R is τI-atomic. Since R{I – Z4, primes must be equivalent to 1, 2, or 3 pmod Iq. Let a P R. Factor a into a unique product of primes a “ p1 ¨ ¨ ¨ pkq1 ¨ ¨ ¨ qlr1 ¨ ¨ ¨ rs where pi ” 1 pmod Iq, qi ” 2 pmod Iq, and ri ” 3 pmod Iq.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Case 1: When a ” 0 or 2 pmod Iq, write a “ q1 ¨ ¨ ¨ ql´1pqlp1 ¨ ¨ ¨ pkr1 ¨ ¨ ¨ rsq for the τI-atomic factorization of a.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Case 1: When a ” 0 or 2 pmod Iq, write a “ q1 ¨ ¨ ¨ ql´1pqlp1 ¨ ¨ ¨ pkr1 ¨ ¨ ¨ rsq for the τI-atomic factorization of a. Case 2: When a ” 1 or 3 pmod Iq, write a “ p1 ¨ ¨ ¨ pkr1 ¨ ¨ ¨ rs “ p´1qsp1 ¨ ¨ ¨ pkp´r1q ¨ ¨ ¨ p´rsq for the τI-atomic factorization of a.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Lemma

Let R be a PID and I an ideal of R. If R{I – Z2rxs{px2 ` xq, then R is τI-atomic.

Lemma

Let R be a PID and I an ideal of R. If R{I – Z2rxs{px2 ` 1q, then R is τI-atomic.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Remark

F4 is the least well behaved with respect to τI-atomicity.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Remark

F4 is the least well behaved with respect to τI-atomicity. We have τI-atomicity, but not under all conditions.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Remark

F4 is the least well behaved with respect to τI-atomicity. We have τI-atomicity, but not under all conditions.

Theorem

Let R be a PID and I be an ideal such that R{I has a unit in every class. Then R is τI-atomic.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Since R is a PID, there is some prime p P R such that I “ ppq.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Since R is a PID, there is some prime p P R such that I “ ppq. Case 1: Assume a ” 0 pmod Iq. Then a “ pkm for some k P N and some m R I and a “ p ¨ ¨ ¨ pppmq is a τI-atomic factorization.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Since R is a PID, there is some prime p P R such that I “ ppq. Case 1: Assume a ” 0 pmod Iq. Then a “ pkm for some k P N and some m R I and a “ p ¨ ¨ ¨ pppmq is a τI-atomic factorization. Case 2: Assume a ı 0 pmod Iq.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Assume a “ p1p2 is a product of primes where pi ı 0 pmod Iq.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Assume a “ p1p2 is a product of primes where pi ı 0 pmod Iq. Since there is a unit in every class, there is some λ with λ ” p1p´1

2

pmod Iq.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Assume a “ p1p2 is a product of primes where pi ı 0 pmod Iq. Since there is a unit in every class, there is some λ with λ ” p1p´1

2

pmod Iq. Then a “ p1p2 “ λ´1p1pλp2q is a τI-atomic factorization of a where λp2 ” p1p´1

2 p2 ” p1

pmod Iq.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

We write a “ p1 ¨ ¨ ¨ pk where each pi is a prime and us the same method to obtain a “ pλ´1

2

¨ ¨ ¨ λ´1

k qp1pλ2p2q ¨ ¨ ¨ pλkpkq.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

We write a “ p1 ¨ ¨ ¨ pk where each pi is a prime and us the same method to obtain a “ pλ´1

2

¨ ¨ ¨ λ´1

k qp1pλ2p2q ¨ ¨ ¨ pλkpkq.

Thus a has a τI-atomic factorization and R is τI-atomic.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Example

Let R “ Zrαs where α “ 1`

? 5 2

and let I “ 2Zrαs. Then R{I “ Zrαs{2Zrαs “ ta ` bα: a, b P Z2u – F4.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Example

Let R “ Zrαs where α “ 1`

? 5 2

and let I “ 2Zrαs. Then R{I “ Zrαs{2Zrαs “ ta ` bα: a, b P Z2u – F4. Then ´1, α and α´1 are all units in R with each one in a different nonzero class.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Lemma

Let R be a PID and I an ideal of R such that R{I has a prime in every class. If R{I – F4 and R does not have a unit in every nonzero class of R{I, then R is not τI-atomic.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Lemma

Let R be a PID and I an ideal of R such that R{I has a prime in every class. If R{I – F4 and R does not have a unit in every nonzero class of R{I, then R is not τI-atomic. Label the four elements of F4 as 0, 1, a, b and create the Cayley table. 0 1 a b 0 0 0 0 0 1 0 1 a b a 0 a b 1 b 0 b 1 a

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

1 a b 1 1 a b a a b 1 b b 1 a

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

1 a b 1 1 a b a a b 1 b b 1 a Since there exists at least one prime in every class, there exists primes p and q with p ” a pmod Iq and q ” b pmod Iq. Then p2q “ pp2qpqq has only one τI-factorization, but p2 is not a τI-atom since p2 “ ppqppq.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

1 a b 1 1 a b a a b 1 b b 1 a Since there exists at least one prime in every class, there exists primes p and q with p ” a pmod Iq and q ” b pmod Iq. Then p2q “ pp2qpqq has only one τI-factorization, but p2 is not a τI-atom since p2 “ ppqppq. Hence R is not τI-atomic.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

The assumption that there is a prime in every class is necessary.

Example

Let R “ F4rrxss and I “ pxq. Then R{I – F4. Recall that any power series with a nonzero constant term is a unit.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

The assumption that there is a prime in every class is necessary.

Example

Let R “ F4rrxss and I “ pxq. Then R{I – F4. Recall that any power series with a nonzero constant term is a unit. Let f P R. Then f “ xng where g has a nonzero constant term.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

The assumption that there is a prime in every class is necessary.

Example

Let R “ F4rrxss and I “ pxq. Then R{I – F4. Recall that any power series with a nonzero constant term is a unit. Let f P R. Then f “ xng where g has a nonzero constant term. This is always a τI-atomic factorization and hence R is τI-atomic.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Theorem

Let R be a PID with a prime in every class and I an ideal of R such that |R{I| “ 4. Then R is τI-atomic if and only if R{I fl F4 and R does not contain a unit in every nonzero class.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Question: Let R “ Z and I “ pnq for some n P Z. Can we determine the τI-elasticity of R for a given n?

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Question: Let R “ Z and I “ pnq for some n P Z. Can we determine the τI-elasticity of R for a given n?

Definition

Let a P R. The τI-elasticity of a, denoted ρτpaq, is the ratio of the longest τI-atomic factorization over the shortest τI-atomic

• factorization. The τI-elasticity of R is ρτpRq “ suptρτpaq|a P Ru.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Recall that if α and 2α ` 1 are both primes, we say α is a Germain prime and 2α ` 1 is a safe prime.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Recall that if α and 2α ` 1 are both primes, we say α is a Germain prime and 2α ` 1 is a safe prime.

Lemma

Let n “ 2α ` 1 be a safe prime. Let p, q and r be primes such that pkpqrq is a τn-factorization. If p ” ˘1 pmod nq, then there exists no other τn-factorization of pkpqrq.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Recall that if α and 2α ` 1 are both primes, we say α is a Germain prime and 2α ` 1 is a safe prime.

Lemma

Let n “ 2α ` 1 be a safe prime. Let p, q and r be primes such that pkpqrq is a τn-factorization. If p ” ˘1 pmod nq, then there exists no other τn-factorization of pkpqrq. If there existed a second τn-factorization it would be of the form ppsqqppk´srq.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Recall that if α and 2α ` 1 are both primes, we say α is a Germain prime and 2α ` 1 is a safe prime.

Lemma

Let n “ 2α ` 1 be a safe prime. Let p, q and r be primes such that pkpqrq is a τn-factorization. If p ” ˘1 pmod nq, then there exists no other τn-factorization of pkpqrq. If there existed a second τn-factorization it would be of the form ppsqqppk´srq. Since p ” ˘1 pmod nq, this would imply q ” ˘r pmod nq. But pqrq cannot be factored since pkpqrq is a τn-factorization.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Theorem

Let n “ 2α ` 1 be a safe prime. Let p be a prime and r be an integer such that r is not equal to ˘1. Then pkr is not a τn-atom for k ě α.

Theorem

Let n “ 2α ` 1 be a safe prime. Then for any integer m we have ρτpmq ď α ´ 1.

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity

Thank you!

Richard Hasenauer, Bethany Kubik τ-factorization and τ-elasticity