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Keywords:
local cohomology module; weakly Laskerian module; ${\mathfrak a}$-weakly Laskerian module; associated prime
local cohomology module; weakly Laskerian module; ${\mathfrak a}$-weakly Laskerian module; associated prime
Summary:
Let $R$ be a commutative Noetherian ring and ${\mathfrak a}$ an ideal of $R$. We introduce the concept of ${\mathfrak a}$-weakly Laskerian $R$-modules, and we show that if $M$ is an ${\mathfrak a}$-weakly Laskerian $R$-module and $s$ is a non-negative integer such that ${\rm Ext}^j_R(R/{\mathfrak a}, H^i_{{\mathfrak a}}(M))$ is ${\mathfrak a}$-weakly Laskerian for all $i<s$ and all $j$, then for any ${\mathfrak a}$-weakly Laskerian submodule $X$ of $H^s_{{\mathfrak a}}(M)$, the $R$-module ${\rm Hom}_R(R/{\mathfrak a},H^s_{{\mathfrak a}}(M)/X)$ is ${\mathfrak a}$-weakly Laskerian. In particular, the set of associated primes of $H^s_{\mathfrak a}(M)/X$ is finite. As a consequence, it follows that if $M$ is a finitely generated $R$-module and $N$ is an ${\mathfrak a}$-weakly Laskerian $R$-module such that $ H^i_{{\mathfrak a}}(N)$ is ${\mathfrak a}$-weakly Laskerian for all $i<s$, then the set of associated primes of $H^s_{\mathfrak a}(M, N)$ is finite. This generalizes the main result of S. Sohrabi Laleh, M. Y. Sadeghi, and M. Hanifi Mostaghim (2012).
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