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.
Abyzov et al. (Sun,) studied this question.