Related papers: Unique factorization property of non-unique factor…

Let $S \subset R$ be an arbitrary subset of a unique factorization domain $R$ and $\K$ be the field of fractions of $R$. The ring of integer-valued polynomials over $S$ is the set $\mathrm{Int}(S,R)= \{ f \in \mathbb{K}[x]: f(a) \in R\…

A subset $S$ of an integral domain $R$ is called a semidomain if the pairs $(S,+)$ and $(S, \cdot)$ are semigroups with identities; additionally, we say that $S$ is additively reduced provided that $S$ contains no additive inverses. Given…

We study quasi-quadratic modules in a pseudo-valuation domain $A$ whose strict units admit a square root. Let $\mathfrak X_R^N$ denote the set of quasi-quadratic modules in an $R$-module $N$, where $R$ is a commutative ring. It is known…

Let H be an algebraic group scheme over a field k acting on a commutative k-algebra A which is a unique factorisation domain. We show that, under certain mild assumptions, the monoid of nonzero H-stable principal ideals in A is free…

Using the notion of cyclically pure injective modules, a characterization for rings which are locally valuation is established. As applications, new characterizations for Prufer domains and pure semi-simple rings are provided. Namely, we…

Let $D$ be a Dedekind domain with infinitely many maximal ideals, all of finite index, and $K$ its quotient field. Let $\operatorname{Int}(D) = \{f\in K[x] \mid f(D) \subseteq D\}$ be the ring of integer-valued polynomials on $D$. Given any…

Local Noetherian domains arising as local rings of points of varieties or in the context of algebraic number theory are analytically unramified, meaning their completions have no nontrivial nilpotent elements. However, looking elsewhere,…

$\DeclareMathOperator{\IntR}{Int{}^\text{R}}$$\DeclareMathOperator{\Int}{Int}$Let $D$ be a domain. Park determined the necessary and sufficient conditions for which the ring of integer-valued polynomials $\Int(D)$ is a globalized…

Let $F$ be a field, let $D$ be a local subring of $F$, and let Val$_F(D)$ be the space of valuation rings of $F$ that dominate $D$. We lift Zariski's connectedness theorem for fibers of a projective morphism to the Zariski-Riemann space of…

An integral domain $D$ is called an irreducible-divisor-finite domain (IDF-domain) if every nonzero element of $D$ has finitely many irreducible divisors up to associates. The study of IDF-domains dates back to the seventies. In this paper,…

Let $V$ be a rank one valuation domain with quotient field $K$. We characterize the subsets $S$ of $V$ for which the ring of integer-valued polynomials ${\rm Int}(S,V)=\{f\in K[X] \mid f(S)\subseteq V\}$ is a Pr\"ufer domain. The…

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…

Let $R$ be a commutative ring with identity. An element $r \in R$ is said to be absolutely irreducible in $R$ if for all natural numbers $n>1$, $r^n$ has essentially only one factorization namely $r^n = r \cdots r$. If $r \in R$ is…

Let $K$ be a characteristic zero algebraic function field with a valuation $\nu$. Let $L$ be a finite extension of $K$ and $\omega$ be an extension of $\nu$ to $L$. We establish that the valuation ring $V_{\omega}$ of $\omega$ is…

In this paper we provide necessary and sufficient conditions for $ R=A\propto E $ to be a valuation ring where $E$ is a non-torsion or finitely generated $A-$module. Also, we investigate the $ (n,d) $ property of the valuation ring.

Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain $(R,M)$ with finite residue field, it is known that there exist absolutely irreducible elements, that is, irreducible elements all of whose…

A ring is *unit-additive* if a sum of units is always either a unit or nilpotent. For example, $k[X]$ and $k[X]/(X^2)$ are unit-additive, but $\mathbb Z$ is not. We prove a wide-ranging theorem about unit-additivity in semigroup rings,…

We classify dp-minimal integral domains, building off the existing classification of dp-minimal fields and dp-minimal valuation rings. We show that if R is a dp-minimal integral domain, then R is a field or a valuation ring or arises from…

Let k be a field. Let K_0(V_k) denote the quotient of the free abelian group generated by the geometrically reduced varieties over k, modulo the relations of the form [X]=[X-Y]+[Y] whenever Y is a closed subvariety of X. Product of…

Unique factorization fails in many rings and monoids, but divisor and transfer homomorphisms provide tools to understand non-unique factorizations. In this expository article, we first explore these notions in the classical setting of…