A NEW INVARIANT DERIVED FROM LOCAL COHOMOLOGY MODULES

Let (R, m) be a commutative Noetherian local ring of dimension d and M a non-zero finitely generated R-module. In this paper, we introduce a new useful invariant for M, denoted by lambda(R)(M). We study some properties of lambda(R)(M) and, especially in case that M ($) over cap is the m-adic completion of M, we find an interval for. lambda(R ($) over cap) (M ($) over cap). It is seen that this invariant inherits some of the properties of Krull dimension on exact sequences. Also, as a nice application of this invariant, it is shown that the set AsshR(R) has only one element in case that lambda(R)(R) = 0 and therefore (0 : H-m(d) (R)) is a primary ideal of R when in addition d = 1.