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On the category of weakly Laskerian cofinite modules. (English) Zbl 1306.13010
Let $$R$$ be a Noetherian commutative ring with identity and $$I$$ an ideal of $$R$$. Recall that an $$R$$-module $$M$$ is said to be $$I$$-cofinite if $$\text{Supp}_RM\subseteq V(I)$$ and $$\text{Ext}_R^i(R/I,M)$$ is finitely generated for all $$i\geq 0$$. Also, $$M$$ is said to be weakly Laskerian if any quotient module of $$M$$ has finitely many associated prime ideals.
The main achievement of this paper is to show that the category of $$I$$-cofinite weakly Laskerian $$R$$-modules forms an abelian subcategory of the category of all $$R$$-modules. Also, the author provides a nice characterization for weakly Laskerian modules. He proves that an $$R$$-module $$M$$ is weakly Laskerian if and only if it possesses a submodule $$N$$ such that $$\text{Supp}_RM/N$$ is finite.

##### MSC:
 13D45 Local cohomology and commutative rings
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