τ-IRREDUCIBLE DIVISOR GRAPHS IN COMMUTATIVE RINGS WITH ZERO-DIVISORS

Year 2016, Volume: 20 Issue: 20, 45 - 69, 01.12.2016

Christopher Park Mooney

https://doi.org/10.24330/ieja.266183

Abstract

In this paper, we combine research done recently in two areas
of factorization theory. The first is the extension of τ-factorization to commutative
rings with zero-divisors. The second is the extension of irreducible
divisor graphs of elements from integral domains to commutative rings with
zero-divisors. We introduce the τ-irreducible divisor graph for various choices
of associate and irreducible. By using τ-irreducible divisor graphs, we find
that we are able to obtain, as subcases, many of the graphs associated with
commutative rings which followed from the landmark 1988 paper by I. Beck.
We then are able to use these graphs to give alternative characterizations of
τ-finite factorization properties previously defined in the literature.

Keywords

Factorization, zero-divisors, commutative rings, zero-divisor graphs, irreducible divisor graphs