A Note on Artinianess of Certain Generalized Local Cohomology Modules

Let ?: R0?R be a ring homomorphism and suppose that a and a0, respectively, are ideals of R and R0 such that is an Artinian ring. Let M and N be two finitely generated R-modules and suppose that (R0,m0) is a local ring. In this note we prove that the R-modules and are Artinian for all integers i and j, whenever and . Also we will show that if a is principal, then the R-modules and are Artinian, for all integers i and j. Moreover, we will show that if is the largest integer i such that is not Artinian, then the R-modules and are Artinian, for and all . Also as a consequence of this result we get that the R-modules and are artinian, for j=d, d?1, in which is the cohomological dimension of M and N with respect to a. Our results recover the corresponding known ones.