Artinian and Non-Artinian Local Cohomology Modules

Abstract Let M be a finite module over a commutative noetherian ring R. For ideals a and b of R, the relations between cohomological dimensions of M with respect to a, b, a ⋂ b and a + b are studied. When R is local, it is shown that M is generalized Cohen–Macaulay if there exists an ideal a such that all local cohomology modules of M with respect to a have finite lengths. Also, when r is an integer such that 0 ≤ r < dimR(M), any maximal element q of the non-empty set of ideals ﹛a : (M) is not artinian for some i, i ≥ r} is a prime ideal, and all Bass numbers of (M) are finite for all i ≥ r.