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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.

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].

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...

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...

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.