Locally Balanced Modules

For an arbitrary left R-module M, a criterion for the existence of an analogue of decomposition of M into primary components is found using M-adic topology and P -adic topology on R, where P is a family of pairwise comaximal ideals from R. A consequence of this criterion is the existence of decomposition into primary components of torsion modules over some rings close to Dedekind rings. A module an analogue of the density theorem holds for is called locally balanced. Locally balanced modules allowing for P -primary decomposition are studied. As applications, the conditions the analogue of the double centralizer theorem holds under for some types of matrix rings are studied.