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 $i \leq cd (𝔞, M) + 1$ , whenever $dim R / 𝔞 = 1$ . In addition, we show that if $M$ is finitely generated and $H_{𝔞}^{i} (M)$ are weakly Laskerian for all $i \leq t - 1$ , then $H_{𝔞}^{i} (M)$ are $𝔞$ -cofinite for all $i \leq t - 1$ 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.

Communicated by E. Gorla

Keywords:

AMSC: 13D45, 14B15, 13E05

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