Related papers: Chains of integrally closed ideals

Let $(R,\fm)$ be a Cohen-Macaulay local ring of positive dimension $d$ and infinite residue field. Let $I$ be an $\fm$-primary ideal of $R$ and $J$ be a minimal reduction of $I$. In this paper we show that if $\widetilde{I^k}=I^k$ and…

Let $(A,\mathfrak m)$ be an excellent two-dimensional normal local domain. In this paper we study the elliptic and the strongly elliptic ideals of $A$ with the aim to characterize elliptic and strongly elliptic singularities, according to…

This paper studies algebraic residual intersections in rings with Serre's condition \( S_{s} \). It demonstrates that residual intersections admit free approaches i.e. perfect subideal with the same radical. This fact leads to determining a…

Let $(A,\mathfrak{m})$ be an analytically unramified Cohen-Macaulay local ring and let $\mathfrak{a}$ be an $\mathfrak{m}$-primary ideal in $A$. If $I$ is an ideal in $A$ then let $I^*$ be the integral closure of $I$ in $A$. Let…

In this note, we define and investigate ideal covering numbers of associative rings (not assumed to be commutative or unital): three invariants defined as the minimal number of proper left, right, or two-sided ideals whose union equals the…

The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a…

We characterize monomial ideals which are intersections of monomial prime ideals and study classes of ideals with this property, among them polymatroidal ideals.

Let $G$ be a complex simple Lie group and let $\g = \hbox{\rm Lie}\,G$. Let $S(\g)$ be the $G$-module of polynomial functions on $\g$ and let $\hbox{\rm Sing}\,\g$ be the closed algebraic cone of singular elements in $\g$. Let ${\cal L}\s…

For the family of graded lattice ideals of dimension 1, we establish a complete intersection criterion in algebraic and geometric terms. In positive characteristic, it is shown that all ideals of this family are binomial set theoretic…

Let $(A,\mathfrak{m})$ be an analytically unramified formally equidimensional Noetherian local ring with $\ depth \ A \geq 2$. Let $I$ be an $\mathfrak{m}$-primary ideal and set $I^*$ to be the integral closure of $I$. Set $G^*(I) =…

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,…

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where "varieties" carry a PGL_n-action, regular…

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is a simple undirected graph whose vertex set is the set of nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if…

Let T be a complete local ring and C a finite set of incomparable prime ideals of T. We find necessary and sufficient conditions for T to be the completion of an integral domain whose generic formal fiber is semilocal with maximal ideals…

We investigate formal power series ideals and their relationship to topological rewriting theory. Since commutative formal power series algebras are Zariski rings, their ideals are closed for the adic topology defined by the maximal ideal…

In the theory of commutative semirings, the lack of additive inverses creates a structural divergence between ideals and congruences that does not exist in ring theory. The aim of this article is to restore critical ideal-theoretic…

Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s)^{\ell} \oplus k(-2s+1)$, where $s \geq3$ and…

The purpose of this note is to revisit the results of arXiv:1407.4324 from a slightly different perspective, outlining how, if the integral closures of a finite set of prime ideals abide the expected convexity patterns, then the existence…

Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided…

We consider the intersection $\mathfrak{M}(A)$ of all maximal ideals of an evolution algebra $A$ and study the structure of the quotient $A/\M(A)$. In a previous work, maximal ideals have been related to hereditary subsets of a graph…