Jim Coykendall
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View article: On the axioms for a unique factorization domain
On the axioms for a unique factorization domain Open
With the growing evolution of the theory of non-unique factorization in integral domains and monoids, the study of several variations to the classical unique factorization domain (or UFD) property have become popular in the literature. Usi…
View article: Overrings of half-factorial orders
Overrings of half-factorial orders Open
The behavior of factorization properties in various ring extensions is a central theme in commutative algebra. Classically, the UFDs are (completely) integrally closed and tend to behave well in standard ring extensions, with the notable e…
View article: Norms, Normsets, and Factorization
Norms, Normsets, and Factorization Open
We present a development of norms and discuss their relationship to factorization. In earlier work, the first named author introduced the notion of a normset, which is the image of the norm map. A normset is a monoid with its own factoriza…
View article: Atomicity in integral domains
Atomicity in integral domains Open
In algebra, atomicity is the study of divisibility by and factorizations into atoms (also called irreducibles). In one side of the spectrum of atomicity we find the antimatter algebraic structures, inside which there are no atoms and, ther…
View article: Square-difference factor absorbing ideals of a commutative ring
Square-difference factor absorbing ideals of a commutative ring Open
Let $R$ be a commutative ring with $1 \neq 0$. A proper ideal $I$ of $R$ is a {\it square-difference factor absorbing ideal} (sdf-absorbing ideal) of $R$ if whenever $a^2 - b^2 \in I$ for $0 \neq a, b \in R$, then $a + b \in I$ or $a - b \…
View article: Adjacency-Like Conditions and Induced Ideal Graphs
Adjacency-Like Conditions and Induced Ideal Graphs Open
In this paper we examine some natural ideal conditions and show how graphs can be defined that give a visualization of these conditions. We examine the interplay between the multiplicative ideal theory and the graph theoretic structure of …
View article: Rings of very strong finite type
Rings of very strong finite type Open
The SFT (for strong finite type) condition was introduced by J. Arnold in the context of studying the condition for formal power series rings to have finite Krull dimension. In the context of commutative rings, the SFT property is a near-N…
View article: Hereditary atomicity in integral domains
Hereditary atomicity in integral domains Open
If every subring of an integral domain is atomic, then we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind d…
View article: On unique factorization domains
On unique factorization domains Open
In this paper we attempt to generalize the notion of “unique factorization domain” in the spirit of “half-factorial domain”. It is shown that this new generalization of UFD implies the now well-known notion of half-factorial domain. As a c…
View article: Length-factoriality in commutative monoids and integral domains
Length-factoriality in commutative monoids and integral domains Open
View article: On the atomicity of monoid algebras
On the atomicity of monoid algebras Open
View article: Spontaneous atomicity for polynomial rings with zero-divisors
Spontaneous atomicity for polynomial rings with zero-divisors Open
In this paper, we show that it is possible for a commutative ring with identity to be non-atomic (that is, there exist non-zero nonunits that cannot be factored into irreducibles) and yet have a strongly atomic polynomial extension. In par…
View article: Elasticity in Polynomial-Type Extensions
Elasticity in Polynomial-Type Extensions Open
The elasticity of an atomic integral domain is, in some sense, a measure of how far the domain is from being a half-factorial domain. We consider the relationship between the elasticity of a domain R and the elasticity of its polynomial ri…