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Endomorphism Rings of Abelian p-Groups

  • Wolfgang Liebert
Part of the Lecture Notes in Mathematics book series (LNM, volume 1006)

Abstract

In [6] we have determined necessary and sufficient conditions that an abstract ring be isomorphic to the ring E(G) of all endomorphisms of an abelian p-group G without elements of infinite height. In the present paper we give a characterization of the endomorphism rings of totally projective abelian p-groups. Combining the methods we used for these characterizations, we are able to characterize the endomorphism rings of a large class of abelian p-groups including all p-groups without elements of infinite height and all totally projective p-groups. A group G belongs to this class if height-increasing homomorphisms from finite subgroups of G into G can always be extended to endomorphisms of G. Our methods require that the groups under consideration have enough cyclic direct summands. Therefore we restrict ourselves to abelian p-groups which are bounded or of infinite length. Our results demonstrate once again that the structure of the ring E(G) is to a large extent determined by the ideal EO(G) consisting of all endomorphisms of G which map G onto a finite subgroup of G. This is not so surprising because for these p-groups, GyH if and only if EO(G) ≅ EO(H).

Keywords

Left Ideal Endomorphism Ring Versus Versus Versus Versus Versus Neighborhood Basis Finite Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • Wolfgang Liebert

There are no affiliations available

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