On pure subrings of sp-groups

Abstract

Let G be a sp-group such that for every prime p, G_p is elementary. We show that ${\text{End}}_{\mathbb{Z}}(G)$ is a sp-group and every subring R of $\prod {\text{End}}_{\mathbb{Z}}(G_p)$, containing $\oplus {\text{End}}_{\mathbb{Z}}(G_p)$ is pure if and only if $R={\mathbb{M}}_T=\{x\in \prod _{p\in \mathbb{P}}\text{End}(G_p)|\exists k\in \mathbb{N}\text{such that} \mathit{\text{kx}}\in T\},$ where T is a subring of $\prod _{p\in \mathbb{P}}\text{End}(G_p)$. We observe that $\frac{{\mathbb{M}}_T}{{\oplus }_{p\in \mathbb{P}}\text{End}(G_p)}$ is (ring) isomorphic with $T{\otimes }_{\mathbb{Z}}\mathbb{Q}$. 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.

Keywords:

• Abelian sp-groups
• endomorphism ring
• pure subring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20K25
• 20K21