相关论文: Complete integral closure and strongly divisorial …

We give conditions for a maximal divisorial ideal to be t-maximal and show with examples that, even in a completely integrally closed domain, maximal divisorial ideals need not be t-maximal.

In this paper we study strongly hollow ideals and completely strongly hollow ideals in commutative rings without finiteness assumptions. We establish basic structural properties, including maximality phenomena and permanence under quotients…

Absolute integral closures of general commutative unital rings are explored. All rings admit absolute integral closures, but in general they are not unique. Among the reduced rings with finitely many minimal prime ideals, finite products of…

There has arisen in recent years a substantial theory of "multiplier ideals'' in commutative rings. These are integrally closed ideals with properties that lend themselves to highly interesting applications. But how special are they among…

An integral domain $D$ is called a \emph{prime-divisor-finite domain} (PDF-domain) if every nonzero element has only finitely many nonassociate prime divisors. A domain $D$ is said to be a \emph{tightly prime-divisor-finite domain}…

Completely prime right ideals are introduced as a one-sided generalization of the concept of a prime ideal in a commutative ring. Some of their basic properties are investigated, pointing out both similarities and differences between these…

We show that for ideals primary to a maximal ideal in a normal domain of finite type over the complex numbers, its tight closure is contained inside the continuous closure.

It is well-known that a class of all modules, which are torsion-free with respect to a set of ideals, is closed under injective envelopes. In this paper, we consider a kind of a dual to this statement - are the divisibility classes closed…

We continue the analysis of prime and semiprime operations over one-dimensional domains started in \cite{Va}. We first show that there are no bounded semiprime operations on the set of fractional ideals of a one-dimensional domain. We then…

Let $S(D)$ represent a set of proper nonzero ideals $I(D)$ (resp., $t$ -ideals $I_{t}(D)$) of an integral domain $D\neq qf(D)$ and let $P$ be a valid property of ideals of $D.$ We say $S(D)$ meets $P$ (denoted $ S(D)\vartriangleleft P)$ if…

In this paper we exhibit an example of a three-dimensional regular local domain (A, n) having a height-two prime ideal P with the property that the extension PA^ of P to the n-adic completion A^ of A is not integrally closed. We use a…

It is a well-known and easily established fact that every Euclidean domain is also a principal ideal domain. However, the converse statement is not true, and this is usually shown by exhibiting as a counterexample the ring of algebraic…

Let $R$ be a commutative ring with identity. For an $R$-module $M$, the notion of strongly prime submodule of $M$ is defined. It is shown that this notion of prime submodule inherits most of the essential properties of the usual notion of…

We study the effects on $D$ of assuming that the power series ring $D[[X]]$ is a $v$-domain or a PVMD. We show that a PVMD $D$ is completely integrally closed if and only if $\bigcap_{n=1}^{\infty }(I^{n})_{v}=(0)$ for every proper…

In this article we revisit a problem regarding Bezout domains, namely, whether every Bezout domain is an elementary divisor domain. We prove that a Bezout domain in which every maximal ideal is principal is an elementary divisor ring

A semiring is uniserial if its ideals are totally ordered by inclusion. First, we show that a semiring $S$ is uniserial if and only if the matrix semiring $M_n(S)$ is uniserial. As a generalization of valuation semirings, we also…

The aim of this series of papers is to study $z$-ideals of semirings. In this article, we introduce some distinguished classes of $z$-ideals of semirings, which include $z$-prime, $z$-semiprime, $z$-irreducible, and $z$-strongly irreducible…

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 define a closure operation for ideals in a commutative ring which has all the good properties of solid closure (at least in the case of equal characteristic) but such that also every ideal in a regular ring is closed. This gives in…

Let $A$ be an integral domain. We study new conditions on families of integral ideals of $A$ in order to get that $A$ is of $t$-finite character (i.e., each nonzero element of $A$ is contained in finitely many $t$-maximal ideals). We also…