相关论文: Integral closure, basically full closure, and dual…
We define a duality operation connecting closure operations, interior operations, and test ideals, and describe how the duality acts on common constructions such as trace, torsion, tight and integral closures, and divisible submodules. This…
Closure operations such as tight and integral closure and test ideals have appeared frequently in the study of commutative algebra. This articles serves as a survey of the authors' prior results connecting closure operations, test ideals,…
This article is a survey of closure operations on ideals in commutative rings, with an emphasis on structural properties and on using tools from one part of the field to analyze structures in another part. The survey is broad enough to…
There are several ways to convert a closure or interior operation to a different operation that has particular desirable properties. In this paper, we axiomatize 3 ways to do so, drawing on disparate examples from the literature, including…
Exploiting the interior-closure duality developed by Epstein and R.G., we show that for the class of Matlis dualizable modules $\mathcal{M}$ over a Noetherian local ring, when cl is a Nakayama closure and i its dual interior, there is a…
We introduce an operation on modules over an $F$-finite ring of characteristic $p$. We call this operation \emph{tight interior}. While it exists more generally, in some cases this operation is equivalent to the Matlis dual of tight…
In this article, we construct integrally closed modules of rank two over a two-dimensional regular local ring. The modules are explicitly constructed from a given complete monomial ideal with respect to a regular system of parameters. Then…
We introduce two closure operations on ideals in commutative rings related to the ring operation of root closure. One closure is the result of iterating a root-like operation on ideals infinitely many times, and the other closure arises as…
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…
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…
We study integrality over rings (all commutative in this paper) and over ideal semifiltrations (a generalization of integrality over ideals). We begin by reproving classical results, such as a version of the "faithful module" criterion for…
We construct a self-dual complete resolution of a module defined by a pair of embedded complete intersection ideals in a local ring. Our construction is based on a gluing construction of Herzog and Martsinkovsky and exploits the structure…
We study certain properties of modules over 1-dimensional local integral domains. First, we examine the order of the conductor ideal and its expected relationship with multiplicity. Next, we investigate the reflexivity of certain…
An integro-differential ring is a differential ring that is closed under an integration operation satisfying the fundamental theorem of calculus. Via the Newton--Leibniz formula, a generalized evaluation is defined in terms of integration…
Let $A\subseteq B$ be a ring extension and $\mathcal{G}$ be a set of $A$-submodules of $B$. We introduce a class of closure operations on $\mathcal{G}$ (which we call \emph{multiplicative operations on $(A,B,\mathcal{G})$}) that generalizes…
The semiring of discrete dynamical systems is a simple algebraic model for modularity in deterministic systems. The objects of the semiring are finite transformations (viewed as directed graphs and regarded up to isomorphism), the sum of…
Optimization of DR-submodular functions has experienced a notable surge in significance in recent times, marking a pivotal development within the domain of non-convex optimization. Motivated by real-world scenarios, some recent works have…
In this paper, we construct indecomposable integrally closed modules of arbitrary rank over a two-dimensional regular local ring. The modules are quite explicitly constructed from a given complete monomial ideal. We also give structural and…
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…
I introduce the concept of integral closure for elements and ideals in idempotent semirings, and establish how it corresponds to its namesake in commutative algebra. In the case of free semirings, integral closure can be understood in terms…