Christopher Park Mooney

• PhD, The University of Iowa
• Professor (Associate) at University of Wisconsin–Stout

In this article, we investigate magic type labelings of zero-divisor graphs. In particular, we turn our attention to semi-magic, magic, and super-magic labelings. We are able to construct infinitely many rings which admit these magic type labelings as well as infinitely many rings which do not have these magic type labeling. We further proceed to c...

We determine necessary and sufficient conditions for broad classes of generalized power series rings to satisfy the ascending chain condition on principal ideals or possess the bounded factorization property. Along the way, we consider when a generalized power series ring is domainlike or (weakly) présimplifiable. As corollaries to our general theo...

We determine necessary and sufficient conditions for broad classes of generalized power series rings to satisfy the ascending chain condition on principal ideals or possess the bounded factorization property. Along the way, we consider when a generalized power series ring is domainlike or (weakly) présimplifiable. As corollaries to our general theo...

We determine necessary and sufficient conditions for broad classes of generalized power series rings to satisfy the ascending chain condition on principal ideals or possess the bounded factorization property. Along the way, we consider when a generalized power series ring is domainlike or (weakly) présimplifiable. As corollaries to our general theo...

Several different versions of "factoriality" have been defined for commutative rings with zero divisors. We apply semigroup theory to study these notions in the context of a commutative monoid ring R[S], determining necessary and sufficient conditions for R[S] to be various kinds of "unique factorization rings." Our work generalizes Anderson et al....

"Unique factorization" was central to the initial development of ideal theory. We update this topic with several new results concerning notions of "unique ideal factorization rings" with zero divisors. Along the way, we obtain new characterizations of several well-known kinds of rings in terms of their ideal factorization properties and examine whe...

In recent decades, mathematicians have extended the definition of a unique factorization domain to rings with zero divisors in many different ways, by mixing and matching different notions of "irreducible" elements, "equivalent" factorizations, and "redundant factorizations" from the literature and deciding which elements are required to have "uniq...

In this article, we study two popular types of vertex labelings of the zero-divisor graph which arises naturally from a commutative ring R with zero-divisors. As a continuation of the early study of zero-divisor graphs about the coloring number, we are instead interested in the graceful labeling of Rosa and the harmonious labeling of Graham and Slo...

"Unique factorization" was central to the initial development of ideal theory. We update this topic with several new results concerning notions of "unique ideal factorization rings" with zero divisors. Along the way, we obtain new characterizations of several well-known kinds of rings in terms of their ideal factorization properties and examine whe...

We study the factorization of ideals of a commutative ring, in the context of the U-factorization framework introduced by Fletcher. This leads to several "U-factorability" properties weaker than unique U-factorization. We characterize these notions, determine the implications between them, and give several examples to illustrate the differences. Fo...

We perform an in-depth study of several different cancellation properties for modules. Among those we consider are (half) (weak) cancellation modules, restricted cancellation modules, and (half) join principal modules. We also investigate which commutative rings have every nonzero (finitely generated) ideal (respectively, module) satisfying some ca...

We study the factorization of ideals of a commutative ring, defining multiple different kinds of "nonfactorable" ideals and several "factorability" properties weaker than unique factorization. We characterize (some of) these notions, determine the implications between them, and give several examples to illustrate the differences. We also examine ho...

In this paper, we combine research done recently in two areas of factorization theory. The first is the extension of τ-factorization to com- mutative 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 gr...

Generalized factorization theory for integral domains was initiated by D. D. Anderson and A. Frazier in 2011 and has received considerable attention in recent years. There has been significant progress made in studying the relation n for the integers in previous undergraduate and graduate research projects. In 2013, the second author extended the g...

Recently, substantial progress has been made on generalized factorization techniques in integral domains, in particular, τ-factorization. There have also been advances made in investigating factorization in commutative rings with zero-divisors. One approach which has been found to be very successful is that of U-factorization introduced by Fletcher...

In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their attention to studying divisor graphs of non-zero elements in the ring, the so called irreducible divisor graph. In thi...

Much work has been done on generalized factorization techniques in integral domains, namely $\tau$-factorization. There has also been substantial progress made in investigating factorization in commutative rings with zero-divisors. There are many ways authors have decided to study factorization when zero-divisors present. This paper focuses on the...

Recently there has been a flurry of research on generalized factorization techniques in both integral domains and rings with zero-divisors, namely $\tau$-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 an...

In 1988, I. Beck introduced the notion of a zero-divisor graph of a commutative rings with $1$. There have been several generalizations in recent years. In particular, in 2007 J. Coykendall and J. Maney developed the irreducible divisor graph. Much work has been done on generalized factorization, especially $\tau$-factorization. The goal of this pa...

Much work has been done on generalized factorization techniques in integral domains, namely τ-factorization. There has also been substantial progress made in investigating factorization in commutative rings with zero-divisors. This paper seeks to synthesize work done in these two areas and extend the notion of τ-factorization to commutative rings t...

The study of factorization in integral domains has a long history. Unique factorization domains, like the integers, have been studied extensively for many years. More recently, mathematicians have turned their attention to generalizations of this such as Dedekind domains or other domains which have weaker factorization properties. Many authors have...