Related papers: Chains of Unusual Excellent Local Rings

Let $(A,{\mathfrak m})$ be a Cohen-Macaulay local ring and let $I$ be an ideal of $A$. We prove that the Rees algebra ${\mathcal R}(I)$ is an almost Gorenstein ring in the following cases: (1) $(A,{\mathfrak m})$ is a two-dimensional…

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…

The primary purpose of this paper is give a classification scheme for the nonzero primes of a Pr\"ufer domain based on five properties. A prime $P$ of a Pr\"ufer domain $R$ could be sharp or not sharp, antesharp or not, divisorial or not,…

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p >0$. Let $\mathcal{M}$ be the thick tensor ideal of finitely generated modules whose support variety is in a fixed subvariety $V$ of the projectivized prime ideal…

In this paper, we prove that the set of all $F$-pure thresholds of ideals with fixed embedding dimension satisfies the ascending chain condition. As a corollary, given an integer $d$, we verify the ascending chain condition for the set of…

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for…

Let $(R, {\mathfrak m})$ be a Noetherian local ring and let $I$ be an ${\mathfrak m}$-primary ideal. In this paper we give sharp bounds on the multiplicity of the special fiber ring ${\mathcal F}$ of $I$ in terms of other well-known…

For each positive prime integer $p$ we construct a standard graded $F$-rational ring $R$, over a field $K$ of characteristic $p$, such that $R\otimes_K\overline{K}$ is not $F$-rational. By localizing we obtain a flat local homomorphism $(R,…

Let $\ast $ be a finite character star operation defined on an integral domain $D.$ Call a nonzero $\ast $-ideal $I$ of finite type a $\ast $ -homogeneous ($\ast $-homog) ideal, if $I\subsetneq D$ and $(J+K)^{\ast }\neq D$ for every pair…

We search for principal ideals. As a sample, let $R$ be a strongly-normal, almost-factorial, and complete-intersection local ring with a prime ideal $P$ of height one. If $depth(R/ P)\geq dim R-2$, we show $P$ is principal. As an immediate…

We generalize the theory of radical factorization from almost Dedekind domain to strongly discrete Pr\"ufer domains; we show that, for a fixed subset $X$ of maximal ideals, the finitely generated ideals with $\mathcal{V}(I)\subseteq X$ have…

The paper is devoted to the study of absolute ideals of groups in the class $\mathcal{QD}1$, which consists of all quotient divisible abelian groups of torsion-free rank 1. A ring is called an $AI$-ring (respectively, an $RF$-ring) if it…

The purpose of this paper is twofold. Firstly, to emphasise that the class of Lie algebras with chain lattices of ideals are elementary blocks in the embedding or decomposition of Lie algebras with finite lattice of ideals. Secondly, to…

Thirty years ago, Huneke (for local rings) and Lyubeznik (in general) conjectured that for all regular rings $R$, the local cohomology modules $H^i_I(R)$ have finitely many associated prime ideals. We prove substantial new cases of their…

A proper ideal $I$ in a commutative ring with unity is called a $z^\circ$-ideal if for each $a$ in $I$, the intersection of all minimal prime ideals in $R$ which contain $a$ is contained in $I$. For any totally ordered field $F$ and a…

Let $(R,\m,k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_\dt$ be a minimal resolution of the residue field $k$, and for each $i\ge 0$, let $\gothic t_i(R) = \lim_{e\to \8}…

Let $R$ be a commutative ring with nonzero identity. A. Yassine et al. defined in the paper (Yassine, Nikmehr and Nikandish, 2020), the concept of $1$-absorbing prime ideals as follows: a proper ideal $I$ of $R$ is said to be a…

An integral domain is called {\em Globalized multiplicatively pinched-Dedekind domain $($GMPD domain$)$} if every nonzero non-invertible ideal can be written as $JP_1\cdots P_k$ with $J$ invertible ideal and $P_1,...,P_k$ distinct ideals…

Let $R$ be a normal Noetherian local domain of Krull dimension two. We examine intersections of rank one discrete valuation rings that birationally dominate $R$. We restrict to the class of prime divisors that dominate $R$ and show that if…

Let $F$ be a field. For each nonempty subset $X$ of the Zariski-Riemann space of valuation rings of $F$, let ${A}(X) = \bigcap_{V \in X}V$ and ${J}(X) = \bigcap_{V \in X}{\mathfrak M}_V$, where ${\mathfrak M}_V$ denotes the maximal ideal of…