Jim Coykendall

Zero-Divisor Graphs of Polynomials and Power Series Over Commutative Rings

ABSTRACT We recall several results about zero-divisor graphs of commutative rings. Then we examine the preservation of diameter and girth of the zero-divisor graph under extension to polynomial and power series rings.

Half-Factorial Domains, a Survey

Let D be an integral domain. D is atomic if every nonzero nonunit of D can be written as a product of irreducible elements (or atoms) of D. Let 1 (D) represent the set of irreducible elements of D. Traditionally, an atomic domain D is a unique factorization domain (UFD) if α 1… α n = β 1… β m for each ai and β j ∈I (D) implies: 1. n =m, 2. there exists a permutation б of {1,... ,n} such that α 1 and β б (i) are associates.

On integral domains with no atoms

Antimatter domains are defined to be the integral domains which do not have any atoms. It is proved that each integral domain can be em-bedded as a subring of some antimatter domain which is not a field. Any fragmented domain is an antimatter domain, but the converse fails in each positive Krull dimension. A detailed study is made of the passage of the“an-timatter”property between the partners within an overring extension. Special attention is given to characterizing antimatter domains in classes of valuation domains, pseudo-valuation domains, and various types of pullbacks.

Irreducible Divisor Graphs

In Beck (1988), the author introduces the idea of a zero-divisor graph of a commutative ring. We generalize this idea to study factorization in integral domains and define irreducible divisor graphs. We use these irreducible divisor graphs to characterize certain classes of domains, including UFDs.

The SFT property does not imply finite dimension for power series rings

Abstract In this paper we give an example to show that if R is finite-dimensional and has the SFT property, then R〚t〛 is not necessarily SFT, nor finite-dimensional.

The half-factorial property in integral extensions

In this paper, the integral closure of a half-factorial subring of a ring of algebraic integers is studied. The boundary map, a natural generalization of the length function of Zaks, is used to show that the integral closure of such an order is again an HFD.

Monoid Domain Constructions of Antimatter Domains

An integral domain without irreducible elements is called an antimatter domain. We give some monoid domain constructions of antimatter domains. Among other things, we show that if D is a GCD domain with quotient field K that is algebraically closed, real closed, or perfect of characteristic p > 0, then the monoid domain D[X; ℚ+] is an antimatter GCD domain. We also show that a GCD domain D is antimatter if and only if P−1 = D for each maximal t-ideal P of D.

Sets with few Intersection Numbers from Singer Subgroup Orbits

Using a Singer cycle in Desarguesian planes of order q? 1(mod3), q a prime power, Brouwer 2 gave a construction of sets such that every line of the plane meets them in one of three possible intersection sizes. These intersection sizes x, y, and z satisfy the system of equationsformulaBrouwer claimed that this system has a unique solution in integers. Further, Brouwer noted that for q a perfect square, this system has a solution for which two of the variables are equal, ostensibly implying that when q is a square the constructed set has only two intersection numbers. In this paper, we perform a detailed analysis which shows that this system does not in general have a unique solution. In particular, the constructed sets when q is a square might have three intersection numbers. The cases for which this occurs are completely determined.

FORMAL POWER SERIES RINGS OVER ZERO-DIMENSIONAL SFT-RINGS

ABSTRACT: Let R be a zero-dimensional SFT-ring. It is proved that the minimal prime ideals of the formal power series ring A=R[[x 1, …, xn ]] are the ideals of the form [[x 1, …, xn ]], where is a (minimal) prime of R. It follows that A has Krull dimension n and is catenarian. If R⊆T where T is also a zero-dimensional SFT-ring, the lying-over, going-up, incomparable, and going-down properties are studied for the extension A⊆T[[x 1, …, xn ]].

On the integral closure of a half-factorial domain

A half-factorial domain (HFD), R, is an atomic integral domain where given any two products of irreducible elements of R: α1α2⋯αn=β1β2⋯βm then n=m. As a natural generalization of unique factorization domains (UFD), one wishes to investigate which “good” properties of UFDs that HFDs possess. In particular, it has been conjectured that the integral closure of a half-factorial domain is again a HFD (see Non-Noetherian Commutative Ring Theory, Mathematics and its applications, Vol. 520, Kluwer, Dordrecht, 2000, pp. 97–115. for example). In this paper we produce an example that demonstrates that the integral closure of a HFD does not even have to be atomic. We shall investigate the failure of this conjecture closely and highlight some cases where the conjecture does indeed hold.