Related papers: A prime ideal principle for two-sided ideals

In this paper we study prime and maximal ideals in a Hurwitz polynomial ring hR. It is well-known that to study many questions we may assume R is prime and consider just R-disjoint ideals. We give a characterization for an R-disjoint ideal…

A well-known result of K\"{o}the and Cohen-Kaplansky states that a commutative ring $R$ has the property that every $R$-module is a direct sum of cyclic modules if and only if $R$ is an Artinian principal ideal ring. This motivated us to…

This article studies the notion of $S-r-$ideals in commutative ring $H$, where $S$ is a multiplicatively closed subset of $H$. Some basic properties of $S-r-$ideals are given. Various characterizations of $S-r-$ideals are presented. Also,…

Let $\ast $ be a star operation of finite character. Call a $\ast $-ideal $I$ of finite type a $\ast $-homogeneous ideal if $I$ is contained in a unique maximal $\ast $-ideal $M=M(I).$ A maximal $\ast $-ideal that contains a $\ast…

In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring (pure spectrum) are obtained. Especially, Grothendieck type theorem is obtained which states that there is a canonical correspondence between…

For $a\in R$, let $P_a$ denote the intersection of all minimal prime ideals of $R$ containing $a$. An ideal $I$ of a ring $R$ is called a $z^{\circ}$-ideal if $P_a\subseteq I$ for all $a\in I$. In this paper, we first investigate the class…

We provide a natural criterion which implies equality of the finitistic test ideal and test ideal in local rings of prime characteristic. Most notably, we show that the criterion is met by every local weakly $F$-regular ring whose…

In this paper we will investigate commutative rings which have the $\ast $-property. We say that a ring $R$ satisfy $\ast-$property if for any family of ideals $\left\{ I_{\alpha}\right\} _{\alpha\in S}$ of $R$ in which $S$ is an index set,…

In a previous paper we exhibited the somewhat surprising property that most direct links of prime ideals in Gorenstein rings are equimultiple ideals with reduction number $1$. This led to the construction of large families of…

Let $R$ be a commutative ring with identity and ${\rm Nil}(R)$ be the set of nilpotent elements of $R$. The nil-graph of ideals of $R$ is defined as the graph $\mathbb{AG}_N(R)$ whose vertex set is $\{I:\ (0)\neq I\lhd R$ and there exists a…

We prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid, and strongly-rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion…

The purpose of this paper is to give an elementary proof to the theorem due to Avramov on certain determinantal ideals of linear type.

A complete understanding of the structure of all prime ideals of an infinite direct product of commutative rings (e.g. in terms of more specific objects) has remained a challenging problem for decades. In this article, new advances have…

Let $(A,\mathfrak{m})$ be an excellent normal domain of dimension two containing a field $k \cong A/\mathfrak{m}$. An $\mathfrak{m}$-primary ideal $I$ to be a $p_g$-ideal if the Rees algebra $A[It]$ is a Cohen-Macaulay normal domain. If $k$…

Let $R$ be a commutative (Noetherian) local ring of prime characteristic $p$ that is $F$-pure. This paper is concerned with comparison of three finite sets of radical ideals of $R$, one of which is only defined in the case when $R$ is…

For which monomial supports do most polynomials generate a prime ideal? We give necessary and sufficient conditions for the radical of the ideal to be prime over an algebraically closed field. In characteristic zero, the same conditions…

We show that if a groupoid graded ring has a certain nonzero ideal property, then the commutant of the center of the principal component of the ring has the ideal intersection property, that is it intersects nontrivially every nonzero ideal…

We begin by investigating the class of commutative unital rings in which no two distinct elements divide the same elements. We prove that this class forms a finitely axiomatizable, relatively ideal distributive quasivariety, and it equals…

We characterize ideals whose adjoints are determined by their Rees valuations. We generalize the notion of a regular system of parameters, and prove that for ideals generated by monomials in such elements, the integral closure and adjoints…

We introduce differential primary decompositions for ideals in a commutative ring. Ideal membership is characterized by differential conditions. The minimal number of conditions needed is the arithmetic multiplicity. Minimal differential…