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On two-sided artinian quotient rings

Published online by Cambridge University Press:  18 May 2009

P. F. Smith
P. F. Smith
Affiliation:
University of Glasgow, Glasgow, G12 8QQ
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Djabali [1] has proved that, if R is a right and left noetherian ring with an identity and if the proper prime ideals of R are maximal, then R has a right and left artinian two-sided quotient ring. Robson [5, Theorem 2.11] and Small [6, Theorem 2.13] have proved independently that, if Ris a commutative noetherian ring, then Rhas an artinian quotient ring if and only if the prime ideals of Rthat belong to the zero ideal are all minimal. We shall generalise these results by proving the

Theorem. Let R be a right and left noetherian ring with a regular element. Then R has a right and left artinian two-sided quotient ring if and only if each prime ideal of R consisting of zero–divisors is minimal.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 13 , Issue 2 , September 1972 , pp. 159 - 163
Copyright
Copyright © Glasgow Mathematical Journal Trust 1972

References

REFERENCES

1.Djabali, M., Anneaux de fractions généralisés artiniens, C. R. Acad. Sci. Paris 268 (1969), 2138–2140.Google Scholar
2.Goldie, A. W., Rings with maximum condition, Mimeographed lecture notes, Yale University, 1961.Google Scholar
3.Goldie, A. W., Lectures on non-commutative noetherian rings, Canad. Math. Congress, York University, Toronto, 1967.Google Scholar
4.Jacobson, N., The theory of rings, Amer. Math. Soc. Surveys No. 2 (New York, 1943).CrossRefGoogle Scholar
5.Robson, J. C., Artinian quotient rings, Proc. London Math. Soc. (3) 17 (1967), 600–616.CrossRefGoogle Scholar
6.Small, L. W., Orders in artinian rings, J. Algebra 4 (1966), 13–41.CrossRefGoogle Scholar
7.Small, L. W., On some questions in noetherian rings, Bull. Amer. Math. Soc. 72 (1966), 853–857.CrossRefGoogle Scholar