Related papers: Periodic modules over Gorenstein local rings

Let $R$ be a commutative noetherian local ring, and let $M$ be a finitely generated $R$-module. Inspired by works of Vasconcelos and Briggs on characterization of complete intersection local rings through the homological properties of the…

Given a finitely generated module $M$ over a commutative local ring (or a standard graded $k$-algebra) $(R,\m,k) $ we detect its complexity in terms of numerical invariants coming from suitable $\m$-stable filtrations $\mathbb{M}$ on $M$.…

Consider an infinite minimal free resolution of a module $M$ over a local Noetherian ring $R$. It was shown by Eisenbud that if $R$ is a complete intersection ring, then a minimal resolution is periodic iff it is bounded. Over more general…

Let $R$ be a commutative Noetherian local ring. We study tensor products involving a finitely generated $R$-module $M$ through the natural action of its endomorphism ring. In particular, we study torsion properties of self tensor products…

Let $T=\left( \begin{array}{cc} R & M 0 & S \end{array} \right) $ be a triangular matrix ring with $R$ and $S$ rings and $_RM_S$ an $R$-$S$-bimodule. We describe Gorenstein projective modules over $T$. In particular, we refine a result of…

Let $R$ be a Noetherian ring, $I$ and $J$ two ideals of $R$ and $t$ an integer. Let $S$ be the class of Artinian $R$-modules, or the class of all $R$-modules $N$ with $\dim_RN\leq k$, where $k$ is an integer. It is proved that $\inf\{i:…

Many known results on finite von Neumann algebras are generalized, by purely algebraic proofs, to a certain class ${\mathcal C}$ of finite Baer *-rings. The results in this paper can also be viewed as a study of the properties of Baer…

If R is a commutative ring, we prove that every finitely generated module has a pure-composition series with indecomposable factors and any two such series are isomorphic if and only if R is a Bezout ring and a CF-ring.

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 ring (not necessarily with 1) and G be a finite group of automorphisms of R. The set B(R, G) of primes p such that p | |G| and R is not p-torsion free, is called the set of bad primes. When the ring is |G|-torsion free, i.e.,…

The study of rings and modules with homological criteria is a cornerstone of commutative algebra. Let $R$ be a commutative Noetherian ring with identity (not necessarily local) and $\frak a$ a proper ideal of $R$. In this paper, a relative…

For a Noetherian regular ring $S$ and for a fixed ideal $J\subset S$, assume that the associated primes of local cohomology module $H^i_J(S)$ does not contain $p$ for some $i\geq 0$, and we call this as a property…

Let $(R,\mm,K)$ be a regular local ring containing a field $k$ such that either char $k=0$ or char $k=p$ and tr-deg $K/\BF_p\geq 1$. Let $g_1,\ldots,g_t$ be regular parameters of $R$ which are linearly independent modulo $\mm^2$. Let…

The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring $R$, called $\mathbb{P}$-radical modules (modules $M$ satisfying the prime radical condition "$(\sqrt[p]{{\cal{P}}M}:M)={\cal{P}}$" for…

A commutative Noetherian ring $R$ is said to be Tor-persistent if, for any finitely generated $R$-module $M$, the vanishing of $\operatorname{Tor}_i^R(M,M)$ for $i\gg 0$ implies $M$ has finite projective dimension. An open question of…

Let $A$ be a Noetherian domain and $R$ be a finitely generated $A$-algebra. We study several features regarding the generic freeness over $A$ of an $R$-module. For an ideal $I \subset R$, we show that the local cohomology modules ${\rm…

Let I be an m-primary ideal of a Noetherian local ring (R,m). We consider the Gorenstein and complete intersection properties of the associated graded ring G(I) and the fiber cone F(I) of I as reflected in their defining ideals as…

A generalization of Grothendieck's non-vanishing theorem is proved for a module which is finite over a local homomorphism. It is also proved that the Gorenstein injective dimension of such a module, if finite, is bounded below by its Krull…

Let $(A,\mathfrak{m})$ be a Gorenstein local ring of dimension $d \geq 1$. Suppose there exists be a non-zero $A$ module $M$ of finite length and finite projective dimension such that $\ell\ell(M)$, the Lowey length of $M$, is equal to…

Let $R$ be a ring. It is proved that $(\mathcal{GP}(R), \mathcal{GP}(R)^\bot)$ is a complete hereditary cotorsion pair, where $\mathcal{GP}(R)$ denotes the class of the Gorenstein projective left $R$-modules. Then we get that each left…