Quotient rings of Noetherian module finite rings

Chatters, A. W.; Hajarnavis, C. R.

doi:10.1090/S0002-9939-1994-1198452-2

Quotient rings of Noetherian module finite rings
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by A. W. Chatters and C. R. Hajarnavis PDF

Proc. Amer. Math. Soc. 121 (1994), 335-341 Request permission

Abstract:

Let R be a Noetherian ring which is a finite module over its centre C. We obtain a necessary and sufficient condition for R to have a full quotient ring. We also show that when such a quotient ring does exist, it is obtained by inverting those elements of C which are regular in R.

References

William D. Blair, Right Noetherian rings integral over their centers, J. Algebra 27 (1973), 187–198. MR 325679, DOI 10.1016/0021-8693(73)90173-7
A. W. Chatters, A non-singular Noetherian ring need not have a classical quotient ring, J. London Math. Soc. (2) 10 (1975), 66–68. MR 364323, DOI 10.1112/jlms/s2-10.1.66
A. W. Chatters and C. R. Hajarnavis, Rings with chain conditions, Research Notes in Mathematics, vol. 44, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980. MR 590045
A. W. Goldie, Some aspects of ring theory, Bull. London Math. Soc. 1 (1969), 129–154. MR 265389, DOI 10.1112/blms/1.2.129
K. R. Goodearl and R. B. Warfield Jr., An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, vol. 16, Cambridge University Press, Cambridge, 1989. MR 1020298
Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
T. Y. Lam, Serre’s conjecture, Lecture Notes in Mathematics, Vol. 635, Springer-Verlag, Berlin-New York, 1978. MR 0485842
J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR 934572
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273

Localisation in noncommutative Noetherian rings

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Lance W. Small, On some questions in Noetherian rings, Bull. Amer. Math. Soc. 72 (1966), 853–857. MR 195902, DOI 10.1090/S0002-9904-1966-11587-2
L. W. Small and J. T. Stafford, Regularity of zero divisors, Proc. London Math. Soc. (3) 44 (1982), no. 3, 405–419. MR 656243, DOI 10.1112/plms/s3-44.3.405
J. T. Stafford, Noetherian full quotient rings, Proc. London Math. Soc. (3) 44 (1982), no. 3, 385–404. MR 656242, DOI 10.1112/plms/s3-44.3.385