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

τ -factorization and τ -elasticity Richard Hasenauer Bethany Kubik 1 Northeastern State University 2 University 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 “ λ b 1 ¨ ¨ ¨ b n is a τ I -factorization of a if λ is a unit, b 1 , . . . , b n are nonzero non-units, and b 1 ” ¨ ¨ ¨ ” b n p mod I q . Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Example Let R “ Z and I “ p 2 q . Then 20 “ 2 ¨ 10 is a τ I -factorization since 2 ” 10 p mod 2 q . Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Example Let R “ Z and I “ p 2 q . Then 20 “ 2 ¨ 10 is a τ I -factorization since 2 ” 10 p mod 2 q . However, 20 “ 4 ¨ 5 is not a τ I -factorization since 4 ı 5 p mod 2 q . Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Example Let R “ Z and I “ p 7 q . Then 30 “ 2 ¨ 3 ¨ 5 “ 6 ¨ 5 “ 2 ¨ 15 “ 3 ¨ 10 . The only valid τ I -factorization of the above list is 3 ¨ 10 since 3 ” 10 p mod 7 q . Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Definition We say a P R is a τ I -atom if, for any τ I -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 τ I -atom if, for any τ I -factorization a “ bc , either b or c is a unit. Definition We say R is τ I -atomic if every nonzero non-unit element has a τ I -factorization into a finite product of τ I -atoms. Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Definition We say a P R is a τ I -atom if, for any τ I -factorization a “ bc , either b or c is a unit. Definition We say R is τ I -atomic if every nonzero non-unit element has a τ I -factorization into a finite product of τ I -atoms. Example Let R “ Z and I “ p 1 q “ Z . Then R is τ I -atomic. Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Example Let R “ Z and I “ p 7 q . Then 44 “ 4 ¨ 11 is a τ I -factorization since 4 ” 11 p mod 7 q . Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Example Let R “ Z and I “ p 7 q . Then 44 “ 4 ¨ 11 is a τ I -factorization since 4 ” 11 p mod 7 q . Also, 4 “ 2 ¨ 2 is a τ I -factorization. Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Example Let R “ Z and I “ p 7 q . Then 44 “ 4 ¨ 11 is a τ I -factorization since 4 ” 11 p mod 7 q . Also, 4 “ 2 ¨ 2 is a τ I -factorization. However, 44 “ 2 ¨ 2 ¨ 11 is not a τ I -factorization. Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Example Let R “ Z and I “ p 7 q . Then 44 “ 4 ¨ 11 is a τ I -factorization since 4 ” 11 p mod 7 q . Also, 4 “ 2 ¨ 2 is a τ I -factorization. However, 44 “ 2 ¨ 2 ¨ 11 is not a τ I -factorization. Since 44 does not factor into a product of τ I -atoms, it follows that R is not τ I -atomic. Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Question: What effect (if any) does the size of R { I have on τ I -factorization? Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Question: What effect (if any) does the size of R { I have on τ I -factorization? Fact If | R { I | “ 2 or 3 , then R is always τ I -atomic. Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Question: What effect (if any) does the size of R { I have on τ I -factorization? Fact If | R { I | “ 2 or 3 , then R is always τ I -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 τ I -factorization? Fact If | R { I | “ 2 or 3 , then R is always τ I -atomic. The first interesting cases occur when | R { I | “ 4. Commutative rings with identity and four elements: Z 2 r x s{p x 2 ` x q , Z 2 r x s{p x 2 ` 1 q . Z 4 , F 4 , 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 “ p a q for some non prime a P R . Thus there is no prime p with p ” 0 p mod I q . Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Lemma Let R be a PID and I an ideal of R. If R { I – Z 4 , then R is τ I -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 – Z 4 , then R is τ I -atomic. Since R { I – Z 4 , primes must be equivalent to 1 , 2 , or 3 p mod I q . Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Lemma Let R be a PID and I an ideal of R. If R { I – Z 4 , then R is τ I -atomic. Since R { I – Z 4 , primes must be equivalent to 1 , 2 , or 3 p mod I q . Let a P R . Factor a into a unique product of primes a “ p 1 ¨ ¨ ¨ p k q 1 ¨ ¨ ¨ q l r 1 ¨ ¨ ¨ r s where p i ” 1 p mod I q , q i ” 2 p mod I q , and r i ” 3 p mod I q . Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Case 1: When a ” 0 or 2 p mod I q , write a “ q 1 ¨ ¨ ¨ q l ´ 1 p q l p 1 ¨ ¨ ¨ p k r 1 ¨ ¨ ¨ r s q for the τ I -atomic factorization of a . Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Case 1: When a ” 0 or 2 p mod I q , write a “ q 1 ¨ ¨ ¨ q l ´ 1 p q l p 1 ¨ ¨ ¨ p k r 1 ¨ ¨ ¨ r s q for the τ I -atomic factorization of a . Case 2: When a ” 1 or 3 p mod I q , write a “ p 1 ¨ ¨ ¨ p k r 1 ¨ ¨ ¨ r s “ p´ 1 q s p 1 ¨ ¨ ¨ p k p´ r 1 q ¨ ¨ ¨ p´ r s q for the τ I -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 – Z 2 r x s{p x 2 ` x q , then R is τ I -atomic. Lemma Let R be a PID and I an ideal of R. If R { I – Z 2 r x s{p x 2 ` 1 q , then R is τ I -atomic. Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Remark F 4 is the least well behaved with respect to τ I -atomicity. Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Remark F 4 is the least well behaved with respect to τ I -atomicity. We have τ I -atomicity, but not under all conditions. Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Remark F 4 is the least well behaved with respect to τ I -atomicity. We have τ I -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 τ I -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 “ p p q . Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Since R is a PID, there is some prime p P R such that I “ p p q . Case 1: Assume a ” 0 p mod I q . Then a “ p k m for some k P N and some m R I and a “ p ¨ ¨ ¨ p p pm q is a τ I -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 “ p p q . Case 1: Assume a ” 0 p mod I q . Then a “ p k m for some k P N and some m R I and a “ p ¨ ¨ ¨ p p pm q is a τ I -atomic factorization. Case 2: Assume a ı 0 p mod I q . Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Assume a “ p 1 p 2 is a product of primes where p i ı 0 p mod I q . Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Assume a “ p 1 p 2 is a product of primes where p i ı 0 p mod I q . Since there is a unit in every class, there is some λ with λ ” p 1 p ´ 1 p mod I q . 2 Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

Assume a “ p 1 p 2 is a product of primes where p i ı 0 p mod I q . Since there is a unit in every class, there is some λ with λ ” p 1 p ´ 1 p mod I q . 2 Then a “ p 1 p 2 “ λ ´ 1 p 1 p λ p 2 q is a τ I -atomic factorization of a where λ p 2 ” p 1 p ´ 1 2 p 2 ” p 1 p mod I q . Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity

We write a “ p 1 ¨ ¨ ¨ p k where each p i is a prime and us the same method to obtain a “ p λ ´ 1 ¨ ¨ ¨ λ ´ 1 k q p 1 p λ 2 p 2 q ¨ ¨ ¨ p λ k p k q . 2 Richard Hasenauer, Bethany Kubik τ -factorization and τ -elasticity