Related papers: Revisiting G-Dedekind domains

Analytic properties of function spaces over the real and the complex fields are different in some ways. This reflects in algebraic properties which are different at times and similar in some other respects. For instance, the ring of…

We study those integral domains in which every proper ideal can be written as an invertible ideal multiplied by a nonempty product of proper radical ideals.

We show that every Dedekind domain $R$ lying between the polynomial rings $\mathbb Z[X]$ and $\mathbb Q[X]$ with the property that its residue fields of prime characteristic are finite fields is equal to a generalized ring of integer-valued…

The *reciprocal complement* $R(D)$ of an integral domain $D$ is the subring of its fraction field generated by the reciprocals of its nonzero elements. Many properties of $R(D)$ are determined when $D$ is a polynomial ring in $n\geq 2$…

It is well-known that an integrally closed domain $D$ can be express as the intersection of its valuation overrings but, if $D$ is not a Pr\"{u}fer domain, the most of valuation overrings of $D$ cannot be seen as localizations of $D$. The…

We study the integral domains D satisfying the following condition: whenever I >AB with I,A,B nonzero ideals, there exist ideals A'>A and B'>B such that I=A'B'.

We prove some results on NIP integral domains, especially those that are Noetherian or have finite dp-rank. If $R$ is an NIP Noetherian domain that is not a field, then $R$ is a semilocal ring of Krull dimension 1, and the fraction field of…

We consider properties of extensions of Krull domains such as flatness that involve behavior of extensions and contractions of prime ideals. Let (R,m) be an excellent normal local domain with field of fractions K, let y be a nonzero element…

Following the terminology introduced by Arnold and Sheldon back in 1975, we say that an integral domain $D$ is a GL-domain if the product of any two primitive polynomials over $D$ is again a primitive polynomial. In this paper, we study the…

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,…

Brewer and Heinzer studied the (integral) domains D having the property that each proper ideal A of D has a comaximal ideal factorization with some additional property. They proved that for a domain D, the following are equivalent: (1) Each…

A commutative integral domain is primary if and only if it is one-dimensional and local. A domain is strongly primary if and only if it is local and each nonzero principal ideal contains a power of the maximal ideal. Hence one-dimensional…

Let A be an integral domain with field of fractions K. We investigate the structure of the overrings B of A (contained in K) that are well-centered on A in the sense that each principal ideal of B is generated by an element of A. We…

Let $D$ be a principal ideal domain and $R(D) = \{\begin{pmatrix} a & b 0 & a \end{pmatrix} \mid a, b \in D\}$ be its self-idealization. It is known that $R(D)$ is a commutative noetherian ring with identity, and hence $R(D)$ is atomic…

We say that a commutative ring R satisfies the restricted minimum (RM) condition if for all essential ideals I in R, factor R/I is an Artinian ring. We will focus on Noetherian reduced rings because in this setting known results for RM…

Let $D$ be an integral domain with quotient field $K$ and let $\mathcal{I}% (D) $ be the set of nonzero ideals of $D$. Call, for $I,J\in \mathcal{I}(D)$ , the product $IJ$ of ideals condensed if $IJ=\{ij|i\in I,j\in J\}.$ Call $D$ a…

Let $F$ be a field, and let Zar$(F)$ be the space of valuation rings of $F$ with respect to the Zariski topology. We prove that if $X$ is a quasicompact set of rank one valuation rings in Zar$(F)$ whose maximal ideals do not intersect to…

Let $(R, \mathfrak{m})$ be a regular local ring of dimension $d \geq 2$. A local monoidal transform of $R$ is a ring of the form $R_1= R[\frac{\mathfrak{p}}{x}]_{\mathfrak{m}_1}$ where $x \in \mathfrak{p}$ is a regular parameter,…

In 2008 N.~Q.~Chinh and P.~H.~Nam characterized principal ideal domains as integral domains that satisfy the follo\-wing two conditions: (i) they are unique factorization domains, and (ii) all maximal ideals in them are principal. We…

Let $\mathfrak D$ be a residually finite Dedekind domain, $a\in \mathfrak D$ be a nonzero element and $\mathfrak n$ be a nonzero ideal of $\mathfrak D$. In this paper we describe the dynamics of the map $x\mapsto ax$ over the quotient ring…