On pure subrings of sp-groups

Abstract

Let $G$ be a sp-group such that for every prime $p$, $G_p$ is elementary. %$\oplus \End_{\zz}(G_p) \leq \End_{\zz}(G) \leq \prod \End_{\zz}(G_p)$. Suppose that $\frac{G}{\oplus_{p\in \mathbb{P}} G_p}$ is torsion-free divisible. %In this article we characterize pure subrings of $\prod_{p\in \mathbb{P}} \End(G_p)$. We show that $\End_{\zz}(G)$ is a sp-group and every subring $R$ of $\prod \End_{\zz}(G_p)$, containing $\oplus \End_{\zz}(G_p)$ is pure if and only if $R=\mathbb{M}_T=\{x\in \prod_{p\in \mathbb{P}}\End(G_p) \;|\; \exists k\in \nn \;\mbox{\rm{such that}} \;\; kx \in T \},$ where $T$ is a subring of $\prod_{p\in \mathbb{P}}\End(G_p)$. We observe that $\frac{\mathbb{M}_T}{\oplus_{p\in \mathbb{P}}\End(G_p)}$ is (ring) isomorphic with $T\otimes_{\zz} \qq$. Moreover, we conclude that a significant number of the examples around the topic can be easily obtained and described by choosing an appropriate subring $T$.