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=1∞Mn with ∑n=1∞Mn 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 Πℵ0Zp which contains the identity and
∑ℵ0Zp. 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).