Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Fully idempotent rings have regular centroids

Author: R. C. Courter
Journal: Proc. Amer. Math. Soc. 43 (1974), 293-296
MSC: Primary 16A32
MathSciNet review: 0330217
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the centroid of a ring all of whose ideals are idempotent is commutative and regular in the sense of von Neumann. The center of a fully idempotent ring is regular. Evidently every regular ring is fully idempotent. One nonregular example is Sasiada's simple radical ring. A subring of the countably infinite row-finite matrices over Sasiada's ring provides an example of a nonsimple, indecomposable, nonregular fully idempotent ring.

References [Enhancements On Off] (What's this?)

  • [1] R. C. Courter, Rings all of whose factor rings are semi-prime, Canad. Math. Bull. 12 (1969), 417-426. MR 40 #7309. MR 0254099 (40:7309)
  • [2] C. Faith, Lectures on injective modules and quotient rings, Lecture Notes in Math., no. 49, Springer-Verlag, Berlin and New York, 1967. MR 37 #2791. MR 0227206 (37:2791)
  • [3] I. Kaplansky, Notes on ring theory, Math. Lecture Notes, University of Chicago, Chicago, Ill., 1957. MR 0427288 (55:322)
  • [4] S. Lajos, On regular duo rings, Proc. Japan Acad. 45 (1969), 157-158. MR 39 #5634. MR 0244319 (39:5634)
  • [5] E. Sasiada, Solution of the problem of existence of a simple radical ring, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 9 (1961), 257. MR 23 #A3157. MR 0125860 (23:A3157)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 16A32

Retrieve articles in all journals with MSC: 16A32

Additional Information

Keywords: Von Neumann regular rings, duo rings, prime rings, semiprime rings, locally matrix rings
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society