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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Indecomposable decompositions and the minimal direct summand containing the nilpotents


Author: G. F. Birkenmeier
Journal: Proc. Amer. Math. Soc. 73 (1979), 11-14
MSC: Primary 16A32
MathSciNet review: 512048
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Abstract: It is well known that an indecomposable right ideal decomposition of a ring is not necessarily unique. In this paper we show that the reduced right ideals of such a decomposition are unique up to isomorphism and the remainder of the decomposition forms the unique MDSN. In the main theorem we use triangular matrices to prove that a ring with an indecomposable decomposition is basically composed of a nilpotent ring, a ring (containing a unity) with an indecomposable decomposition which equals its MDSN, and a direct sum of indecomposable reduced rings with unity.


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

  • [1] G. F. Birkenmeier, A decomposition theory of rings, Ph.D. Thesis, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin, 1975.
  • [2] G. F. Birkenmeier, Self-injective rings and the minimal direct summand containing the nilpotents, Comm. Algebra 4 (1976), no. 8, 705–721. MR 0419526 (54 #7547)
  • [3] Joachim Lambek, Lectures on rings and modules, With an appendix by Ian G. Connell, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966. MR 0206032 (34 #5857)
  • [4] B. L. Osofsky, A remark on the Krull-Schmidt-Azumaya theorem, Canad. Math. Bull. 13 (1970), 501–505. MR 0274518 (43 #281)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0512048-6
PII: S 0002-9939(1979)0512048-6
Keywords: Nilpotent elements, reduced ring
Article copyright: © Copyright 1979 American Mathematical Society