On bounded Artinian-finitary modules

Let G be a group, R be a ring and A be an RG-module. We say that A is an Artinian-finitary module over RG if for every element g is an element of G, the factor-module A/C-A(g) is an Artinian R-module. The study of these modules was initiated by Wehrfritz. If D is a Dedekind domain and U is an Artinian D-module, then we can associate with U some numerical invariants. If V is the maximal divisible submodule of U, then V is a direct sum of finitely many indecomposable submodules. The number b(d)( U) of these direct summands is an invariant of U. The composition length b(F) (U/V) of U/V is another invariant of U. We consider the following special case of Artinian-finitary modules. Let D be a Dedekind domain and G be a group. The DG-module A is said to be bounded Artinian-finitary, if A is Artinian-finitary and there are the numbers bF ( A) = b, b(d)(A) = d is an element of N and a finite subset b(sigma)(A) = tau Spec( D) such that l(F) (A/C-A(g)) <= b, l(d)(A/C-A(g)) <= d and Ass(D)(A/C-A(g)) subset of tau for every element g is an element of G. In the article, the bounded Artinian-finitary modules under some natural restriction are studied.