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A Shorter Proof of Goldie's Theorem

Published online by Cambridge University Press:  20 November 2018

Julius Zelmanowitz
Julius Zelmanowitz*
Affiliation:
University of California, Santa Barbara
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In this note we present an extremely short proof of Goldie's theorem on the structure of semiprime Noetherian rings [1]. The outline of the proof was given by Procesi and Small in [4]. By utilizing the concept of the singular ideal of a ring we have been able to weaken the hypotheses of many of the steps in [4]. Most significantly, we are able to avoid a reduction to the case of prime rings, and in Lemma 5 we give an informative list of the relationship between regular elements and essential ideals of semiprime rings.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 5 , October 1969 , pp. 597 - 602
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Goldie, A. W., Semiprime rings with maximum condition, Proc. Lond. Math. Soc. 10 (1960) 201–220.Google Scholar
2. Jacobson, Z. N., Structure of rings (revised edition). Vol. 37, A.M. S. Colloq. (Providence, 1964).Google Scholar
3. Johnson, R. E. and Wong, E. T., Self-injective rings. Canad. Math. Bull. 2 (1959) 167–173.Google Scholar
4. Procesi, C. and Small, L., On a theorem of Goldie. Jour. Algebre 2 (1965) 80–84.Google Scholar