On pure subrings of sp-groups

2024/01/15 by Amini, A., Amini, B., Momtahan, E. · 1 citation
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2401.07742

Abstract

Let G be a sp-group such that for every prime p, G_p is elementary. %⊕ \End_\zz(G_p) ≤ \End_\zz(G) ≤ ∏ \End_\zz(G_p). Suppose that \fracG⊕_{p∈ ℙ} G_p is torsion-free divisible. %In this article we characterize pure subrings of ∏_{p∈ ℙ} \End(G_p). We show that \End_\zz(G) is a sp-group and every subring R of ∏ \End_\zz(G_p), containing ⊕ \End_\zz(G_p) is pure if and only if R=\mathbbM_T=\x∈ ∏_{p∈ ℙ}\End(G_p) | ∃ k∈ \nn \mbox\rmsuch that kx ∈ T \, where T is a subring of ∏_{p∈ ℙ}\End(G_p). We observe that \frac\mathbbM_T⊕_{p∈ ℙ}\End(G_p) is (ring) isomorphic with T⊗_\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.

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