On the endomorphism ring of the canonical module

Journal of Algebra and Its Applications, 2012

Let R be a commutative Noetherian ring and let I be an ideal of R. In this paper, we study the amalgamated duplication ring R I which is introduced by D'Anna and Fontana. It is shown that if R satisfies Serre's condition (Sn) and Ip is a maximal Cohen-Macaulay Rp -module for every p ∈ Spec (R), then R I satisfies Serre's condition (Sn). Moreover if R I satisfies Serre's condition (Sn), then so does R. This gives a generalization of the same result for Cohen-Macaulay rings in [D'Anna, A construction of Gorenstein rings, J. Algebra 306 (2006) 507-519]. In addition it is shown that if R is a local ring and Ann R (I) = 0, then R I is quasi-Gorenstein if and only if b R satisfies Serre's condition (S 2 ) and I is a canonical ideal of R. This result improves the result of D'Anna which is corrected by Shapiro and states that if R is a Cohen-Macaulay local ring, then R I is Gorenstein if and only if the canonical ideal of R exists and is isomorphic to I, provided Ann R (I) = 0.

Communications in Algebra, 2003

For a ring S, let K 0 (FGFl(S)) and K 0 (FGPr(S)) denote the Grothendieck groups of the category of all finitely generated flat S-modules and the category of all finitely generated projective S-modules respectively. We prove that a semilocal ring R is semiperfect if and only if the group homomorphism K 0 (FGFl(R)) → K 0 (FGFl(R/J(R))) is an epimorphism and K 0 (FGFl(R)) = K 0 (FGPr(R)).

Proceedings of the Japan Academy, Series A, Mathematical Sciences, 1987

Introduction. Let R be a fixed (not necessarily commutative) ring. Throughout this nte, we are concerned with left R-modules M, A, H, Like in'Goldie [1], we shall use the following terminology. A non-zero submodule K of M is called essential in M (or M is an essential extension

arXiv: Commutative Algebra, 2017

The notion of generalized Gorenstein local ring (GGL ring for short) is one of the generalizations of Gorenstein rings. In this article, there is given a characterization of GGL rings in terms of their canonical ideals and related invariants.

Let $I$ denote an ideal of a local ring $(R,\mathfrak{m})$ of dimension $n$. Let $M$ denote a finitely generated $R$-module. We study the endomorphism ring of the local cohomology module $H^c_I(M), c = \grade (I,M)$. In particular there is a natural homomorphism $\Hom_{\hat{R}^I}(\hat{M}^I, \hat{M}^I)\to \Hom_{R}(H^c_{I}(M),H^c_{I}(M))$, where $\hat{\cdot}^I$ denotes the $I$-adic completion functor. We prove sufficient conditions such that it becomes an isomorphism. Moreover, we study a homomorphism of two such endomorphism rings of local cohomology modules for two ideals $J \subset I$ with the property $\grade(I,M) = \grade(J,M)$. Our results extends constructions known in the case of $M = R$ (see e.g. \cite{h1}, \cite{p7}, \cite{p1}).

Bulletin of the Australian Mathematical Society

It is shown that a von Neumann regular ring R is left seif-injective if and only if every finitely generated torsion-free left R-module is projective. It is further shown that a countable self-injective strongly regular ring is Artin semi-simple.

Journal of the Australian Mathematical Society, 2017

Our main purpose is to extend several results of interest that have been proved for modules over integral domains to modules over arbitrary commutative rings $R$ with identity. The classical ring of quotients $Q$ of $R$ will play the role of the field of quotients when zero-divisors are present. After discussing torsion-freeness and divisibility (Sections 2–3), we study Matlis-cotorsion modules and their roles in two category equivalences (Sections 4–5). These equivalences are established via the same functors as in the domain case, but instead of injective direct sums $\oplus Q$ one has to take the full subcategory of $Q$-modules into consideration. Finally, we prove results on Matlis rings, i.e. on rings for which $Q$ has projective dimension $1$ (Theorem 6.4).

2012

Let (R;m) be a Noetherian local ring which is a homomorphic image of a local Gorenstein ring and let M be a nitely generated R-module of dimension d > 0. According to Schenzel (Sc1), M is called a Cohen-Macaulay canonical module (CMC module for short) if the canonical module K(M) of M is Cohen-Macaulay. We give another characterization of CMC modules. We describe the non Cohen-Macaulay canonical locus nCMC(M) of M. If d6 4 then nCMC(M) is closed in Spec(R). For each d 5 there are reduced geometric local rings R of dimension d such that nCMC(R) is not stable under specialization 1 .

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry

Let R be a commutative unital ring and a ∈ R. We introduce and study properties of a functor aΓ a (−), called the locally nilradical on the category of R-modules. aΓ a (−) is a generalisation of both the torsion functor (also called section functor) and Baer's lower nilradical for modules. Several local-global properties of the functor aΓ a (−) are established. As an application, results about reduced R-modules are obtained and hitherto unknown ring theoretic radicals as well as structural theorems are deduced.