Endomorphism Rings of Abelian p-Groups
In  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).
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