An Investigation of the Structure Underlying Irreducible Devisors

Abstract

In previous literature Coykendall & Maney, as well as Axtell & Stickles, have discussed the concept of irreducible divisor graphs of elements in domains and ring with zero-divisors respectively, with two different definitions. In this paper we seek to look at the irreducible divisor graphs of ring elements under a hybrid definition of the two previous ones—in hopes that this graph will reveal structure concerning irreducible divisors in rings with zero-divisors. We also compare the three graphs and examine in what respects they are related. Other graph-theoretic properties of this graph will also be discussed.

Irreducible divisor simplicial complexes

“…Since their introduction, irreducible divisor graphs have been studied in the context of integral domains [Axtell et al 2011;Maney 2008] and in more general contexts [Axtell and Stickles 2008;Bachman et al 2012;Smallwood and Swartz 2009;2008]. Despite the appealing result mentioned above, it is difficult to pick out the factorizations of an element given its irreducible divisor graph.…”

Untitled

“…Since their introduction, irreducible divisor graphs have been studied in the context of integral domains [Axtell et al 2011;Maney 2008] and in more general contexts [Axtell and Stickles 2008;Bachman et al 2012;Smallwood and Swartz 2009;. Despite the appealing result mentioned above, it is difficult to pick out the factorizations of an element given its irreducible divisor graph.…”