Let (R,m) be a commutative Noetherian local ring the maximal ideal m
and A an Artinian R-module with Ndim A = d. For each system of parameters x =
(x1, ...,xd) of A, we denote by e(x,A) the multipility of A with respect to x. Let n =
(n1, n2, ...,nd) be a d-tuple of positive integers. The paper concerns to the function of
d-variables
I(x(n);A) := R(0 :A (xn1
1 , ..., xnd
d )R) − e(xn1
1 , ...,xnd
d ;A),
where R(−) is the length of function. We show in this paper that this function may be
not a polynomial in the general case, but the least degree of all upper-bound polynomials
for the function is a numerical invariant of A. A characterization for co Cohen-Macaulay
modules in term of this new invariant is also given.
