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The principal ideal theorem in prime Noetherian rings

Published online by Cambridge University Press:  18 May 2009

A. W. Chatters  and
M. P. Gilchrist
A. W. Chatters
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW
M. P. Gilchrist
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT
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In the study of certain prime Noetherian rings it is natural to consider the set C of elements which are regular modulo all height-1 prime ideals of R. For R commutative, this set C is simply the set of units. In general this is not the case, though with certain additional conditions we can state non-commutative versions of the Principal Ideal Theorem.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 28 , Issue 1 , January 1986 , pp. 63 - 68
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

REFERENCES

1.Amitsur, S. A. and Small, L. W., Prime ideals in P.I. rings, J. Algebra 62 (1980), 358–383.Google Scholar
2.Chatters, A. W., Goldie, A. W., Hajarnavis, C. R. and Lenagan, T. H., Reduced rank in Noetherian rings, J. Algebra 61 (1979), 582–589.Google Scholar
3.Chatters, A. W. and Hajarnavis, C. R., Rings with chain conditions (Pitman, 1980).Google Scholar
4.Cohn, P. M., Algebra, volume 2 (Wiley, 1977).Google Scholar
5.Formanek, E., Halpin, P. and Li, W.-C. W., The Poincaré series of the ring of 2 × 2 generic matrices, J. Algebra 69 (1981), 105–112.Google Scholar
6.Hajarnavis, C. R. and Williams, S., Maximal orders in Artinian rings, J. Algebra 90 (1984), 375–384.Google Scholar
7.Jategaonkar, A. V., Relative Krull dimension and prime ideals in right Noetherian rings, Comm. Algebra 2 (1974), 429–468.Google Scholar
8.Jategaonkar, A. V., Jacobson's conjecture and modules over fully bounded Noetherian rings, J. Algebra 30 (1974), 103–121.Google Scholar
9.Jategaonkar, A. V., Principal ideal theorem for Noetherian P.I. rings, J. Algebra 35 (1975), 17–22.Google Scholar
10.Krause, G., Lenagan, T. H. and Stafford, J. T., Ideal invariance and Artinian quotient rings, J. Algebra 55 (1978), 145–154.Google Scholar
11.Maury, G. and Raynaud, J., Ordres maximaux au sens de K. Asano, Lecture Notes in Mathematics No. 808 (Springer, 1980).Google Scholar
12.Resco, R., Small, L. W. and Stafford, J. T., Krull and global dimensions of semiprime Noetherian PI-rings, Trans. Amer. Math. Soc. 274 (1982), 285–295.Google Scholar
13.Robson, J. C., Idealisers and hereditary Noetherian prime rings, J. Algebra 22 (1975), 45–81.Google Scholar
14.Robson, J. C., Artinian quotient rings, Proc. London Math. Soc. (3) 17 (1967), 600–616.Google Scholar
15.Small, L. W. and Stafford, J. T., Homological properties of generic matrix rings, Israel J. Math., to appear.Google Scholar
16.Smith, P. F., Localisation and the AR property, Proc. London Math. Soc. (3) 22 (1971), 39–68.Google Scholar
17.Stafford, J. T., Stable structure of non-commutative Noetherian rings, J. Algebra 47 (1977), 244–267.Google Scholar