Decompositions of Abelian $p$-groups

Abstract:

Using some elementary properties of endomorphism rings and their radical ideals, an equivalence between the category of $p$-rings and the category of Boolean rings and some examples introduced by the author, it is shown that for every countable atomic Boolean algebra there is a $p$-group without elements of infinite height, standard basic subgroup and no proper isomorphic subgroups which contains a maximal lattice of summands isomorphic to the given Boolean algebra. Moreover, it is established that this lattice is representative in the sense that it determines, up to isomorphism, all the summands of the group.