Related papers: Multiplicities of Indecomposable Injectives

Let $G$ be a real reductive Lie group, $L$ a compact subgroup, and $\pi$ an irreducible admissible representation of $G$. In this article we prove a necessary and sufficient condition for the finiteness of the multiplicities of $L$-types…

The existence of the Gorenstein projective precovers over arbitrary rings is an open question. It is known that if the ring has finite Gorenstein global dimension, then every module has a Gorenstein projective precover. We prove here a…

We survey noetherian rings $A$ over which the injective hull of every simple module is locally artinian. Then we give a general construction for algebras $A$ that do not have this property. In characteristic 0, we also complete the…

Let $(K,M)$ be a pair satisfying some mild condition, where $K$ is a class of $R$-modules and $M$ is a class of $R$-homomorphisms. We show that if $f:A\rightarrow B$ and $g:B\rightarrow A$ are $M$-embeddings and $A,B$ are $K_M$-injective,…

Let $(A, \frak m)$ be a noetherian local ring with maximal ideal $\frak{m}$ and infinite residue field $k = A/\frak{m}.$ Let $J$ be an $\frak m$-primary ideal, $I_1,...,I_s$ ideals of $A$, and $M$ a finitely generated $A$-module. In this…

It is proved that given any prime ideal $\mathfrak{p}$ of height at least 2 in a countable commutative noetherian ring $A$, there are uncountably many more dualizable objects in the $\mathfrak{p}$-local $\mathfrak{p}$-torsion stratum of the…

We describe here some recent progress pertaining to the Serre Intersection Multiplicity Conjecture. In particular, we show that if A is an unramified regular local ring, then just as in the equicharacteristic case, the intersection…

We show that every integrally closed $\mathfrak{m}$-primary ideal $I$ in a commutative Noetherian local ring $(R,\mathfrak{m},k)$ has maximal complexity and curvature, i.e., $ {\rm cx}_R(I) = {\rm cx}_R(k) $ and $ {\rm curv}_R(I) = {\rm…

Given an abelian category, we introduce a categorical concept of (strongly) Gorenstein projective (resp., injective) objects, by defining a new special class of objects. Then we study the transfer of these properties when passing to an…

A torsion theoretical characterization of left Noetherian rings $R$ over which injective hulls of simple left modules are locally Artinian is given. Sufficient conditions for a left Noetherian ring to satisfy this finiteness condition are…

It has recently been shown that a minimal reversible nonsymmetric ring has order 256 answering a questioned original posed in a paper on a taxonomy of 2-primal rings. Answers to similar questions on minimal rings relating to this taxonomy…

Let $M$ and $N$ be modules over a commutative ring $R$ with $N$ Noetherian. We define the injective capacity of $M$ with respect to $N$ over $R$ to be the supremum of the values $t$ for which $N^{\oplus t}$ embeds into $M$. In a dual…

For a noetherian ring R we call an R-module M cofinite if there exists an ideal I of R such that M is I-cofinite; we show that every cofinite module M satisfies dim_R(M)<=injdimR(M). As an application we study the question which local…

In 2012, Ananthnarayan, Avramov and Moore give a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. Given a Gorenstein ring, one would like to know whether it decomposes as a connected sum and…

Let R be a graded ring and n > 1 an integer. In this paper, We introduce the notions of Ding n-gr-injective and Ding n-gr-flat modules by using of special finitely presented graded modules . Then, some properties of Ding n-gr-injective and…

Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base…

Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no non-zero injective semimodules (e.g. the semiring of non-negative integers). In this…

In the paper, we focus on the silting properties and the combinatorial properties of silting and Gorenstein, which is called Gorenstein silting, where the main tools used are recollements of module categories and tensor products. For a ring…

We study homological properties of test modules that are, in principle, modules that detect finite homological dimensions. The main outcome of our results is a generalization of a classical theorem of Auslander and Bridger: we prove that,…

Let $R$ be a ring and $S$ a multiplicative subset of $R$. In this note, we study the localization of $S$-injective modules and $u$-$S$-injective modules under $S$-Noetherian rings and $u$-$S$-Noetherian rings, respectively. The…