... denote a commutative Noetherian local ring with the max-imal ideal m and A an Artinian R-module with Ndim A = d>0. For each system ofparameters x=(x1, ,xd)forA, we denote by e(x; A) the ... Because A is Artinian R-module, it can beconsidered as anR-module. Note that for each element a ∈ R and each element x ∈ A, we can see that ax and ax are the same, where a is the image ... UniversityAbstract.Let (R, m) be a commutative Noetherian local ring the maximal ideal mand A an Artinian R-module with Ndim A = d. For each system of parameters x =(x1, ,xd) of A, we denote...