Artinianness of Local Cohomology Modules Defined by a Pair of Ideals

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 dimR/(I+J)=0, then H^iI,J(M)/JH^iI,J(M) is I-cofinite Artinian for all i≥0; let dimRM/JM=d (2) if R is local and S is a non-zero Serre subcategory of the category of R-modules satisfying the condition CI, then HdI,J(M)/ JHdI,J(M)∈S (3) if M has finite Krull dimension, then H^d+1I,J(M)/JH^d+1I,J(M)=0. Furthermore, notion of (I,J)-relative Goldie dimension of modules is defined and it is shown that H^nI,J(M)/JH^nI,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.