PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 102, Number 4, April 1988 ON THE JACOBSON RADICAL OF SOME ENDOMORPHISM RINGS MANFRED DUGAS (Communicated by Bhama Srinivasan) ABSTRACT. In this note we deal with a question raised by R. S. Pierce in 1963: Determine the elements of the Jacobson radical of the endomorphism ring of a primary abelian group by their action on the group. We concentrate on separable abelian p-groups and give a counterexample to a conjecture of A. D. Sands. We also show that the radical can be pinned down if the endomorphism ring is a split-extension of its ideal of all small maps. Introduction. All groups in this note are abelian p-groups for some fixed but arbitrary prime p. Our notations are standard as in [F]. It is known that the endomorphism ring End(A) of an abelian p-group A determines the group up to isomorphism. R. Pierce [P] raised the question of describing the Jacobson radical J(End(A)) of End(A) by its action on the group. This problem was solved by W. Liebert [L], J. Hausen [H] and Hausen-Johnson [HJ] for S-cyclic, torsion- complete and sufficiently projective p-groups. (For a separable p-group sufficiently projective is the same as wi-separable.) If A is a (separable) p-group, let H(A) = {ip E End(A)| |x| < \xip\ for all 0 ^ x E A[p}} be the ideal of all maps acting height increasing on the socle of A, and let C(A) be the ideal of all elements of End(A) mapping each Cauchy sequence in A[p] onto a convergent one. (For x E A, \x\ denotes the p-height of x in A and topological notations refer to the p-adic topology.) If A is torsion-complete, J(End(A)) = H(A), if A is E-cyclic or in- separable, J(End(A)) = H(A) n C(A), and H(A) n (7(A) C J(End(A)) for all separable p-groups (cf. [S]). The purpose of this paper is to show that J(End(A)) is in general not equal to C(A) n H(A) for separable p-groups A. We will use that J(End(A)) n ES(A) = ES(A) n H (A), where ES(A) is the ideal of all small endomorphisms of A (cf. [S]). Recently, many complicated p-groups have been constructed in [DG1, DG2, CG]. All these groups enjoy the property that End(A) is a split extension of ES(A), i.e. End(A) = R®Ea(A) for some subring R of End(A). The way these groups are constructed, R n H (A) = pR and Hr(A) = HR(A), i.e. if r E R —H(A), then for all n there is 0 ^ x E pnA[p\ such that x and xr have the same height. In this situation Theorem 1 below implies J(End(A)) = (J(R) n H(A)) © (ES(A) n H(A)) and we have J(End(A)) = H(A) f\C(A) for these groups. We will construct a ring R and use the realization result in [C] to obtain a separable p-group A such that Received by the editors July 17, 1986 and, in revised form, December 18, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 20K10, 20K30. Key words and phrases. Abelian p-group, endomorphism ring, small endomorphisms, Jacobson radical, height-preserving maps. ©1988 American Mathematical Society 0002-9939/88 Î1.00 + $.25 per page 823 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use