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

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