Primary modules determined by indecomposable idempotent endomorphisms.

Abstract

A faithful primary module over a complete discrete valuation ring is determined up to isomorphism by any subring of the endomorphism ring of the module which contains all the indecomposable indempotent endomorphisms. This note is part of a study centered on the problem of ascertaining to what extent the decompositions of a primary Abelian group determine the group. Perhaps the most useful interpretation of this problem lies in the connection between direct summands of a group and idempotent elements in the endomorphism ring of the group. One of the advantages of this viewpoint is that it facilitates the introduction of Boolean structures into the general theory of Abelian p-groups (see, for example, [5]). This becomes important once consideration is given to the state of the search for invariants for primary Abelian groups. In short, there seems to be no other recourse but to redefine, in terms of structural acceptability, what the invariants may be. It is the main objective of this note to show that there is some basis for the belief that the theory of Boolean rings can serve as an important vehicle for the characterization of Abelian p-groups. I. Kaplansky has shown that faithful primary modules over complete discrete valuation rings are determined up to isomorphism by their endomorphism rings ([2, Theorem 28]). The fact that there are proper subrings of the endomorphism ring which determine the module was established by R. S. Pierce (see [4, 12.2 and 3.9]). Using the definition of indecomposable idempotent found in [3, p. 132], and employing some modifications, Kaplansky's proof can easily be extended to validate the following theorem. THEOREM. Let R be a complete discrete valuation ring, M and N faithful primary R-modules. If E and F are, respectively, R-subrings of the Received by the editors December 15, 1970 and, in revised form, March 8, 1971. AMS 1970 subject classifications. Primary 20K10, 20K25; Secondary 20K30.