Most abelianp-groups are determined by the Jacobson radical of their endomorphism rings

Abstract

A celebrated theorem due to Baer and Kaplansky states that two abelian p-groups are isomorphic if (and only if) their endomorphism rings are isomorphic [B, K]. In fact, every isomorphism between their endomorphism rings is induced by an isomorphism between the groups. In this note we show that an abelian p-group G is already determined by an ideal, K(G), which is contained in the Jacobson radical J(~(G)) of its endomorphism ring o~(G), provided G has an unbounded basic subgroup. Let K (G) denote the set of all torsion elements of J(g(G)). Then K(G) is a two-sided ideal of ~(G) which we shall call the torsion radical of g(G). The torsion radical is zero when G is a divisible or an elementary abelian p-group. Thus, K(G) does not, in general, reveal the structure of G. However, for p-groups which are unbounded modulo their maximal divisible subgroup, we have the following result.