Commutative rings with two-absorbing factorization

Abstract

Abstract. We use the concept of 2-absorbing ideal introduced by Badawi to study those commutative rings in which every proper ideal is a product of 2-absorbing ideals (we call them TAF-rings). Any TAF-ring has dimension at most one and the local TAF-domains are the atomic pseudo-valuation domains.

On n-absorbing ideals of locally divided commutative rings

“…On the other hand, Mukhtar et.al. [52] considered rings in which each proper ideal is a finite product of 2-absorbing ideals, and called such rings TAF-rings. It follows that every general ZPI-ring is a TAF-ring.…”

“…For instance, let X be an indeterminate and F a field that is not algebraically closed. If L is an algebraic closure of F , then R = F + XL[X] is a TAF-domain [52,Corollary 4.8], but R is not integrally closed. On the other hand, we have the following.…”

“…As we have seen from Proposition 19, TAF-domains have some interesting ring-theoretic properties. On the other hand, in [52], the authors focus on Noetherian TAF-domains. In particular, they characterized when a Noetherian domain R with R : R ′ = (0) is a TAF-domain [52,Corollary 4.10].…”

“…On the other hand, in [52], the authors focus on Noetherian TAF-domains. In particular, they characterized when a Noetherian domain R with R : R ′ = (0) is a TAF-domain [52,Corollary 4.10]. Note that if R is a Noetherian domain, then R is Mori and R ′ = R * .…”

Commutative rings with one-absorbing factorization

“…An ideal I of R is called an n-absorbing ideal of R, if whenever a 1 , a 2 , :::, a nþ1 2 R and Q nþ1 i¼1 a i 2 I, then there are n of the a i 's whose product is in I. In this case, due to Choi and Walker [13,Theorem 1], ð ffiffi I p Þ n I: In [23], Mukhtar et al studied the commutative rings whose ideals have a TA-factorization. A proper ideal is called a TA-ideal if it is a 2-absorbing ideal.…”