On automorphism groups and endomorphism rings of abelian $p$-groups

Abstract:

Let $A$ be a noncyclic abelian $p$-group where $p \geqslant 5$, and let ${p^\infty }A$ be the maximal divisible subgroup of $A$. It is shown that $A/{p^\infty }A$ is bounded and nonzero if and only if the automorphism group of $A$ contains a minimal noncentral normal subgroup. This leads to the following connection between the ideal structure of certain rings and the normal structure of their groups of units: if the noncommutative ring $R$ is isomorphic to the full ring of endomorphisms of an abelian $p$-group, $p \geqslant 5$, then $R$ contains minimal twosided ideals if and only if the group of units of $R$ contains minimal noncentral normal subgroups.