Related papers: Arf rings, simple singularities and reflexive modu…
We prove a tight connection between reflexive modules over a one-dimensional ring $R$ and its birational extensions that are self-dual as $R$-modules. Consequently, we show that a complete local reduced Arf ring has finitely many…
Finitely generated reflexive modules over commutative Noetherian rings form a key component of Auslander and Bridger's stable module theory and are likewise essential in the study of Cohen--Macaulay representations. Recently, H. Dao…
We provide a certain direct-sum decomposition of reflexive modules over (one-dimensional) Arf local rings. We also see the equivalence of three notions, say, integrally closed ideals, trace ideals, and reflexive modules of rank one (i.e.,…
Noetherian rings have played a fundamental role in commutative algebra, algebraic number theory, and algebraic geometry. Along with their dual, Artinian rings, they have many generalizations, including the notions of isonoetherian and…
Matlis duality for modules over commutative rings gives rise to the notion of Matlis reflexivity. It is shown that Matlis reflexive modules form a Krull-Schmidt category. For noetherian rings the absence of infinite direct sums is a…
We study reflexive ideals in one-dimensional Cohen-Macaulay local rings, providing characterizations of almost Gorenstein rings, rings with minimal multiplicity, and Arf rings, which describe their reflexive fractional ideals.
Let R be a Cohen-Macaulay local ring possessing a canonical module. In this paper we consider when the maximal ideal of R is self-dual, i.e. it is isomorphic to its canonical dual as an R-module. local rings satisfying this condition are…
A notion of rigidity with respect to an arbitrary semidualizing complex C over a commutative noetherian ring R is introduced and studied. One of the main result characterizes C-rigid complexes. Specialized to the case when C is the relative…
We establish a link between trace modules and rigidity in modules over Noetherian rings. Using the theory of trace ideals we make partial progress on a question of Dao, and on the Auslander-Reiten conjecture over Artinian Gorenstein rings.
In this note, finite modules locally of finite injective dimension over commutative Noetherian rings are characterized in terms of vanishing of Ext modules.
This paper mainly focuses on commutative local domains of dimension one. We then obtain a criterion for a ring to have a finite number of trace ideals in terms of integrally closed ideals. We also explore properties of such rings related to…
We study rings which have Noetherian cohomology under the action of a ring of cohomology operators. The main result is a criterion for a complex of modules over such a ring to have finite injective dimension. This criterion generalizes, by…
We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective…
In this note, some properties of finitely generated two-periodic modules over commutative Noetherian local rings have been studied. We show that under certain assumptions on a pair of modules $\left(M,N \right)$ with $M$ two-periodic, the…
Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and $A^\circ$ is pure in $R^A$, then the categories of rational left $A$-modules and right $A^\circ$-comodules are isomorphic. In the Hopf algebra case, we can also…
Let $R$ be a commutative noetherian local ring. In this paper, we study the self-duality and eventual periodicity of minimal free resolutions of finitely generated $R$-modules in terms of their syzygy modules and Ext modules. As an…
A commutative ring $R$ is stable provided every ideal of $R$ containing a nonzerodivisor is projective as a module over its ring of endomorphisms. The class of stable rings includes the one-dimensional local Cohen-Macaulay rings of…
In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T.…
In this paper, we aim to obtain some results under the condition that the dual of a module over a commutative Noetherian ring has finite Gorenstein dimension. In this direction, we derive results involving vanishing of Ext as well as the…
Let M, N be free modules over a Noetherian commutative ring R and let F be a field such that card(F) does not exceed the continuum. Then : (1) The assertion that [Any two F-vector spaces with isomorphic duals are isomorphic] is equivallent…