The Pierce decomposition and Pierce embedding of endomorphism rings of abelian p-groups

Abstract

Abstract The first goal of this paper is to investigate the Pierce decomposition of the endomorphism ring End ⁡ ( G ) = F ^ ⊕ End s ⁡ ( G ) {\operatorname{End}(G)=\widehat{F}\oplus\operatorname{End}_{s}(G)} of an abelian p-group G and its application to the recent studies of groups with minimal full inertia and of thick-thin groups. The second goal is to investigate the Pierce embedding Ψ : End ⁡ ( G ) / H ⁢ ( G ) → ∏ n M f n ⁢ ( G ) . \Psi:\operatorname{End}(G)/H(G)\to\prod_{n}M_{f_{n}(G)}. We prove that more classes of groups than those described by Pierce have the property that the map Ψ is surjective, and we furnish examples of groups which do not have this property. Several results connecting the Pierce decomposition and the Pierce embedding of End ⁡ ( G ) {\operatorname{End}(G)} are obtained that allow one to derive general conditions on a group G which ensure that the Pierce embedding of End ⁡ ( G ) {\operatorname{End}(G)} is not surjective.