Vol. 20, No. 3, 1967

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Endomorphism rings of primary abelian groups

Robert William Stringall

Vol. 20 (1967), No. 3, 535–557

This paper is concerned with the study of certain homomorphic images of the endomorphism rings of primary abelian groups. Let E(G) denote the endomorphism ring of the abelian p-group G, and define H(G) = {α E(G)|x G,px = 0 and height x < imply height α(x) > height x}. Then H(G) is a two sided ideal in E(G) which always contains the Jacobson radical. It is known that the factor ring E(G)∕H(G) is naturally isomorphic to a subring R of a direct product Πn=1Mn with n=1Mn contained in R and where each Mn is the ring of all linear transformations of a Zp space whose dimension is equal to the n 1 Ulm invarient of G. The main result of this paper provides a partial answer to the unsolved question of which rings R can be realized as E(G)∕H(G).

Theorem. Let R be a countable subring of Π0Zp which contains the identity and 0Zp. Then there exists a p. group G with a standard basic subgroup and containing no elements of infinite height such that E(G)∕H(G) is isomorphic to R. Moreover, G can be chosen without proper isomorphic subgroups; in this case, H(G) is the Jacobson radical of E(G).

Mathematical Subject Classification
Primary: 20.30
Received: 20 January 1966
Published: 1 March 1967
Robert William Stringall