$\tau $-Regular factorization in commutative rings with zero-divisors

Abstract

Abstract.Recently there has been a flurry of research on generalized factorization techniques in both integral domains and rings with zero-divisors, namely τ -factorization. There are several ways that authors have studied factorization in rings with zero-divisors. This paper focuses on the method of regular factorizations introduced by D.D. Anderson and S. Valdes-Leon. We investigate how one can extend the notion of τ -factorization to commutative rings with zerodivisors by using the regular factorization approach. The study of regular factorization is particularly effective because the distinct notions of associate and irreducible coincide for regular elements. We also note that the popular U-factorization developed by C.R. Fletcher also coincides since every regular divisor is essential. This will greatly simplify many of the cumbersome finite factorization definitions that exist in the literature when studying factorization in rings with zero-divisors. 2010

Τ-Irreducible DIVISOR GRAPHS IN COMMUTATIVE RINGS WITH ZERO-DIVISORS

“…We also find that as in [27], when we deal with regular elements of a commutative ring with zero-divisors, the notions of irreducible and associate will also coincide.…”

“…Every divisor of a regular element is regular and hence non-zero. Thus for any divisor a of x, we have a is τ -irreducible ⇔ a is τ -strongly irreducible ⇔ a is τ -m-irreducible ⇔ a is τ -unrefinably irreducible ⇔ a is τ -very strongly irreducible as in [27]. Thus all the types of irreducible coincide as well as the associate relations, thus each graph has the same vertex set and edge set.…”

“…By using τ -factorization, the authors were able to consolidate much of the research in factorization in integral domains into a single study. Recently, the author was able to extend many of these generalized factorization techniques in rings with zero-divisors in [21,23,24,25,27]. In [22], the author extended irreducible divisor graphs by way of τ -factorization in domains.…”

Τ -U-Factorizations AND THEIR GRAPHS IN COMMUTATIVE RINGS WITH ZERO-DIVISORS

“…With this last result in mind, if we restrict factorization to regular elements as in [18], then all of the results from Proposition 2.17 hold. That is,…”

$\tau$-Atomicity and Quotients of Size Four

“…The first appearance in the literature in on τ -factorization can be found in [3]. It is further studied in [2,4,5,8,10]. In this paper, we ask, given any PID, what is the smallest quotient such that R/I fails to be τ I -atomic.…”