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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Fully idempotent rings have regular centroids
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by R. C. Courter PDF
Proc. Amer. Math. Soc. 43 (1974), 293-296 Request permission

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
  • R. C. Courter, Rings all of whose factor rings are semi-prime, Canad. Math. Bull. 12 (1969), 417–426. MR 254099, DOI 10.4153/CMB-1969-052-2
  • Carl Faith, Lectures on injective modules and quotient rings, Lecture Notes in Mathematics, No. 49, Springer-Verlag, Berlin-New York, 1967. MR 0227206
  • Irving Kaplansky, Topics in commutative ring theory, University of Chicago, Department of Mathematics, Chicago, Ill., 1974. Lecture notes. MR 0427288
  • Sándor Lajos, On regular duo rings, Proc. Japan Acad. 45 (1969), 157–158. MR 244319
  • E. Sąsiada, Solution of the problem of existence of simple radical ring, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 9 (1961), 257 (English, with Russian summary). MR 125860
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 43 (1974), 293-296
  • MSC: Primary 16A32
  • DOI: https://doi.org/10.1090/S0002-9939-1974-0330217-0
  • MathSciNet review: 0330217