Uniserial Groups

Shalom Feigelstock

Abstract: An abelian group $G$ is $E$-cyclic (uniserial) if $G$ is a cyclic (uniserial) module over its endomorphism ring $E(G)$. In this note, abelian groups $G$ which are uniserial $R$-modules, for $R$ a subring of $E(G)$, are studied. Results of J. Hausen on $E$-uniserial groups are generalized to $R$-uniserial groups. It will be shown that if $G$ is a finite rank reduced torsion free group, $R$ is a commutative subring of $E(G)$, satisfying $RG=G$, and $G$ is $R$-uniserial, then $R$ is a domain whose lattice of ideals is totally ordered, and $G$ is the additive group of a ring isomorphic to $R$.

Classification (MSC2000): 20K15.

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