Narrow Results

The aim of this paper is to prove that the study of lifting and weak lifting modules over commutative noetherian rings reduces to the case of coatomic modules, radical artinian Kirby-p-coprimary modules and radical minimax Kirby-p-coprimary modules with over local rings.

The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal $𝔞$ of a Noetherian ring $R$. It is shown that an $R$-module $M$ is cofinite with respect to $𝔞$, if and only if, ${Ext}_R^i(R/𝔞,M)$ is finitely generated for all $i\le \mathrm{\text{cd}}(𝔞,M)+1$, whenever $dimR/𝔞=1$. In addition, we show that if $M$ is finitely generated and $H_{𝔞}^i(M)$ are weakly Laskerian for all $i\le t-1$, then $H_{𝔞}^i(M)$ are $𝔞$-cofinite for all $i\le t-1$ and for any minimax submodule $K$ of $H_{𝔞}^t(M)$, the $R$-modules ${Hom}_R(R/𝔞,H_{𝔞}^t(M)/K)$ and ${Ext}_R^1(R/𝔞,H_{𝔞}^t(M)/K)$ are finitely generated, where $t$ is a non-negative integer. Finally, we explore a criterion for weakly cofiniteness of modules with respect to an ideal of dimension one. Namely, for such ideals it suffices that the two first $Ext$-modules in the definition for weakly cofiniteness are weakly Laskerian. As an application of this result, we deduce that the category of all $𝔞$-weakly cofinite modules over $R$ forms a full Abelian subcategory of the category of modules.

Let $R$ be a commutative Noetherian ring, $𝔞$ an ideal of $R$ and $M$ an $R$-module with $dimM=d$. We get equivalent conditions for top local cohomology module ${\mathrm{\text{H}}}_{𝔞}^d(M)$ to be Artinian and $𝔞$-cofinite Artinian separately. In addition, we prove that if $(R,𝔪)$ is a local ring such that ${\mathrm{\text{Ext}}}_R^i(R/𝔞,M)$ is minimax, for each $i\le d$, then ${\mathrm{\text{Ext}}}_R^i(N,M)$ is minimax $R$-module for each $i\ge 0$ and for each finitely generated $R$-module $N$ with $\mathrm{\text{dim}}N\le 2$ and ${\mathrm{\text{Supp}}}_R(N)\subseteq V(𝔞)$. As a consequence we prove that if $\mathrm{\text{dim}}R/𝔞=2$ and ${\mathrm{\text{Supp}}}_R(M)\subseteq \mathrm{\text{V}}(𝔞)$, then $M$ is $𝔞$-cominimax if (and only if) ${\mathrm{\text{Hom}}}_R(R/𝔞,M)$, ${\mathrm{\text{Ext}}}_R^1(R/𝔞,M)$ and ${\mathrm{\text{Ext}}}_R^2(R/𝔞,M)$ are minimax. We also prove that if $dimR/𝔞=2$ and $n\in {\mathbb{N}}_0$ such that ${\mathrm{\text{Ext}}}_R^i(R/𝔞,M)$ is minimax for all $i\le n+1$, then ${\mathrm{\text{H}}}_{𝔞}^i(M)$ is $𝔞$-cominimax for all $i<n$ if (and only if) ${\mathrm{\text{Hom}}}_R(R/𝔞,{\mathrm{\text{H}}}_{𝔞}^i(M))$ is minimax for all $i\le n$.

Let R be a commutative Noetherian ring, I, J two ideals of R, and M an R-module. For a non-negative integer t, we show: (a) If for all j < t and are finite (resp., Artinian), then is finite (resp., Artinian). (b) If for all j < t and are finite (resp., Artinian), then is finite (resp., Artinian). In addition, if (R,𝔪) is a local ring, J a non-nilpotent ideal, and M a finite R-module, then we show that is not Artinian for some i ∈ ℕ₀.