On the Jacobson radical of some endomorphism rings

Dugas, Manfred

doi:10.1090/S0002-9939-1988-0934850-4

On the Jacobson radical of some endomorphism rings
HTML articles powered by AMS MathViewer

by Manfred Dugas PDF

Proc. Amer. Math. Soc. 102 (1988), 823-826 Request permission

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.

References

A. L. S. Corner, On endomorphism rings of primary abelian groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 277–296. MR 258949, DOI 10.1093/qmath/20.1.277
A. L. S. Corner and Rüdiger Göbel, Prescribing endomorphism algebras, a unified treatment, Proc. London Math. Soc. (3) 50 (1985), no. 3, 447–479. MR 779399, DOI 10.1112/plms/s3-50.3.447
Manfred Dugas and Rüdiger Göbel, On endomorphism rings of primary abelian groups, Math. Ann. 261 (1982), no. 3, 359–385. MR 679796, DOI 10.1007/BF01455456

Almost

-cycle abelian

-groups

László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
Jutta Hausen, Quasi-regular ideals of some endomorphism rings, Illinois J. Math. 21 (1977), no. 4, 845–851. MR 468743
Jutta Hausen and Johnny A. Johnson, Ideals and radicals of some endomorphism rings, Pacific J. Math. 74 (1978), no. 2, 365–372. MR 480615
Wolfgang Liebert, The Jacobson radical of some endomorphism rings, J. Reine Angew. Math. 262(263) (1973), 166–170. MR 325665, DOI 10.1515/crll.1973.262-263.166
J. M. Irwin and E. A. Walker (eds.), Topics in Abelian groups, Scott, Foresman & Co., Chicago, Ill., 1963. MR 0168630
A. D. Sands, On the radical of the endomorphism ring of a primary abelian group, Abelian groups and modules (Udine, 1984) CISM Courses and Lect., vol. 287, Springer, Vienna, 1984, pp. 305–314. MR 789827, DOI 10.1007/978-3-7091-2814-5_{2}2