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Volume 35 • Number 4 • 2012
• Artinianness of Local Cohomology Modules Defined by a Pair of Ideals
Sh. Payrovi and M. Lotfi Parsa

Let $R$ be a commutative Noetherian ring and $I$, $J$ two ideals of $R$. Let $M$ be a finitely generated $R$-module; it is shown that (1) if $\dim R/(I+J)=0$, then $H^{i}_{I,J}(M)/{JH^{i}_{I,J}(M)}$ is $I$-cofinite Artinian for all $i\geq 0$; let $\dim_{R} M/JM=d$ (2) if $R$ is local and $S$ is a non-zero Serre subcategory of the category of $R$-modules satisfying the condition $C_I$, then $H^{d}_{I,J}(M)/$ ${JH^{d}_{I,J}(M)}\in S$ (3) if $M$ has finite Krull dimension, then $H^{d+1}_{I,J}(M)/{JH^{d+1}_{I,J}(M)}=0$. Furthermore, notion of $(I,J)$-relative Goldie dimension of modules is defined and it is shown that $H^{n}_{I,J}(M)/{JH^{n}_{I,J}(M)}$ is Artinian, whenever $M$ is a $ZD$-module of dimension $n$ such that the $(I,J)$-relative Goldie dimension of any quotient of $M$ is finite.

2010 Mathematics Subject Classification: 13D45, 14B15, 13E10.

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