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The existence of relative pure injective envelopes

Fatemeh Zareh-Khoshchehreh; Kamran Divaani-Aazar — 2013

Colloquium Mathematicae

Let 𝓢 be a class of finitely presented R-modules such that R∈ 𝓢 and 𝓢 has a subset 𝓢* with the property that for any U∈ 𝓢 there is a U*∈ 𝓢* with U* ≅ U. We show that the class of 𝓢-pure injective R-modules is preenveloping. As an application, we deduce that the left global 𝓢-pure projective dimension of R is equal to its left global 𝓢-pure injective dimension. As our main result, we prove that, in fact, the class of 𝓢-pure injective R-modules is enveloping.

Cofiniteness of generalized local cohomology modules

Kamran Divaani-Aazar; Reza Sazeedeh — 2004

Colloquium Mathematicae

Let denote an ideal of a commutative Noetherian ring R, and M and N two finitely generated R-modules with pd M < ∞. It is shown that if either is principal, or R is complete local and is a prime ideal with dim R/ = 1, then the generalized local cohomology module $H_{}^{i} (M, N)$ is -cofinite for all i ≥ 0. This provides an affirmative answer to a question proposed in [13].

A criterion for rings which are locally valuation rings

Kamran Divaani-Aazar; Mohammad Ali Esmkhani; Massoud Tousi — 2009

Colloquium Mathematicae

Using the notion of cyclically pure injective modules, a characterization of rings which are locally valuation rings is established. As applications, new characterizations of Prüfer domains and pure semisimple rings are provided. Namely, we show that a domain R is Prüfer if and only if two of the three classes of pure injective, cyclically pure injective and RD-injective modules are equal. Also, we prove that a commutative ring R is pure semisimple if and only if every R-module is cyclically pure...

Pretty cleanness and filter-regular sequences

Somayeh Bandari; Kamran Divaani-Aazar; Ali Soleyman Jahan — 2014

Czechoslovak Mathematical Journal

Let $K$ be a field and $S = K [x_{1}, ..., x_{n}]$ . Let $I$ be a monomial ideal of $S$ and $u_{1}, ..., u_{r}$ be monomials in $S$ . We prove that if $u_{1}, ..., u_{r}$ form a filter-regular sequence on $S / I$ , then $S / I$ is pretty clean if and only if $S / (I, u_{1}, ..., u_{r})$ is pretty clean. Also, we show that if $u_{1}, ..., u_{r}$ form a filter-regular sequence on $S / I$ , then Stanley’s conjecture is true for $S / I$ if and only if it is true for $S / (I, u_{1}, ..., u_{r})$ . Finally, we prove that if $u_{1}, ..., u_{r}$ is a minimal set of generators for $I$ which form either a $d$ -sequence, proper sequence or strong $s$ -sequence (with respect to the reverse lexicographic...

On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence

Kosar Abolfath Beigi; Kamran Divaani-Aazar; Massoud Tousi — 2022

Czechoslovak Mathematical Journal

Let $R$ be a local ring and $C$ a semidualizing module of $R$ . We investigate the behavior of certain classes of generalized Cohen-Macaulay $R$ -modules under the Foxby equivalence between the Auslander and Bass classes with respect to $C$ . In particular, we show that generalized Cohen-Macaulay $R$ -modules are invariant under this equivalence and if $M$ is a finitely generated $R$ -module in the Auslander class with respect to $C$ such that $C \otimes_{R} M$ is surjective Buchsbaum, then $M$ is also surjective Buchsbaum.

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Currently displaying 1 – 5 of 5

The existence of relative pure injective envelopes

Cofiniteness of generalized local cohomology modules

A criterion for rings which are locally valuation rings

Pretty cleanness and filter-regular sequences

On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence