Abelian groups whose endomorphism rings are V-rings

Abstract

We study Abelian groups whose endomorphism rings are V-rings. Let $G$ be a non-reduced Abelian group, We prove that $\mathrm{\text{End}}(G)$ is a V-ring on either side if and only if $G=B\oplus {\mathbb{Q}}^n$ where $B$ is a tame elementary Abelian group. We observe that a reduced group whose endomorphism is a V-ring, is an sp-group. Recognizing that $\mathrm{\text{End}}(G)$ is also an sp-group of ${\prod }_{p\in \mathbb{P}}\mathrm{\text{End}}(G_p)$, we show that $\mathrm{\text{End}}(G)/\oplus \mathrm{\text{End}}(G_p)$ is a V-ring if and only if $\mathrm{\text{End}}(G)$ is a V-ring.