Related papers: Integral closure of 1-dimensional rings

Let $R$ be a commutative Noetherian local ring. Assume that $R$ has a pair $\{x,y\}$ of exact zerodivisors such that $\dim R/(x,y)\ge2$ and all totally reflexive $R/(x)$-modules are free. We show that the first and second Brauer--Thrall…

Our purpose in this work is multifold. First, we provide general criteria for the finiteness of the projective and injective dimensions of a finite module $M$ over a (commutative) Noetherian ring $R$. Second, in the other direction, we…

Let $\{ R_n, {\mathfrak m}_n \}_{n \ge 0}$ be an infinite sequence of regular local rings with $R_{n+1}$ birationally dominating $R_n$ and ${\mathfrak m}_nR_{n+1}$ a principal ideal of $R_{n+1}$ for each $n$. We examine properties of the…

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 $n\geq 2$…

In this work we describe the local cohomology of reflexive modules of rank one over normal semigroup rings with respect to monomial ideals. Using our description we show that the problem of classifying maximal Cohen-Macaulay modules of rank…

We investigate when the Rees algebra of an integrally closed $\mathfrak{m}$-primary ideal in a regular local ring is a Cohen-Macaulay normal domain. While this property always holds in dimension two, it fails in general in higher…

We prove that the integral closedness of any ideal of height at least two is compatible with specialization by a generic element. This opens the possibility for proofs using induction on the height of an ideal. Also, with additional…

The main focus of this paper is on determining the highest non-vanishing local cohomology modules of \Omega_(B/R), \Omega_(B/V)(\Omega_(B/k)) where R is either a complete regular local ring or a complete local normal domain with coefficient…

Among the finitely generated modules over a Noetherian ring R, the semidualizing modules have been singled out due to their particularly nice duality properties. When R is a normal domain, we exhibit a natural inclusion of the set of…

In a formally unmixed Noetherian local ring, if the colength and multiplicity of an integrally closed ideal agree, then $R$ is regular. We deduce this using the relationship between multiplicity and various ideal closure operations.

We call a right module $M$ (strongly) virtually regular if every (finitely generated) cyclic submodule is isomorphic to a direct summand. $M$ is said to be completely virtually regular if every submodule is virtually regular. In this paper,…

Let R be a commutative noetherian ring. In this paper, we study specialization-closed subsets of Spec R. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of…

Let R be an integral domain and I a nonzero ideal of R. A sub-ideal J of I is a t-reduction of I if (JI^{n})_{t}=(I^{n+1})_{t} for some positive integer n. An element x in R is t-integral over I if there is an equation x^{n} + a_{1}x^{n-1}…

In this paper, we study Noetherian local rings $R$ having a finite number of trace ideals. We proved that such rings are of dimension at most two. Furthermore, if the integral closure of $R/H$, where $H$ is the zeroth local cohomology, is…

We consider rings whose one-sided ideals are close to automorphism-invariant modules. We study rings in which every (finitely generated) right ideal is automorphism invariant and rings in which every right ideal is a finite direct sum of…

Recently, many authors have embraced the study of certain properties of modules such as projectivity, injectivity and flatness from an alternative point of view. Rather than saying a module has a certain property or not, each module is…

We study reflexive modules over one dimensional Cohen-Macaulay rings. Our key technique exploits the concept of $I$-Ulrich modules.

We develop the basic properties of an essentially new closure operation on submodules, the \emph{liftable integral closure} of a submodule, including its relationships with the two prevailing notions of integral closure of submodules. We…

Let $(R,{\bf m})$ be a two-dimensional regular local ring with infinite residue field. We prove an analogue of the Hoskin-Deligne length formula for a finitely generated, torsion-free, integrally closed $R$-module $M$. As a consequence, we…

Let $R$ be a domain of Krull dimension one, we study when the class $\mathcal{F}$ of modules over $R$ that are arbitrary direct sums of finitely generated torsion-free modules is closed under direct summands. If $R$ is local, we show that…