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Krull dimension and invertible ideals in Noetherian rings

Published online by Cambridge University Press:  20 January 2009

T. H. Lenagan
T. H. Lenagan
Affiliation:
Mathematical Institute, University of Edinburgh
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In this note we consider the question: If R is a right Noetherian ring and I is an invertible ideal of R, how do the Krull dimensions of various modules, factor rings and over-rings of R, connected with I, compare with the Krull dimension of R? This question is prompted by results in (5) and (6). In comparing the Krull dimension of the ring R with that of the ring R/I, the best result would be that the Krull dimension of the ring R is exactly one greater than that of the ring R/I. This result is not true in general; however, we see, in Theorem 2.4, that if the invertible ideal is contained in the Jacobson radical the result holds. In the general case we find it is necessary to introduce an over-ring T of R generated by the inverse I−1 of I. We then see that the Krull dimension of R is the larger of two possibilities: (a) Krull dimension of R/I plus one or (b) Krull dimension of T. In order to prove this result we construct a strictly increasing map from the poset of right ideals of R to the cartesian product of the poset of right ideals of T with a poset of certain infinite sequences of right ideals of R/I.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 20 , Issue 2 , September 1976 , pp. 81 - 86
Copyright
Copyright © Edinburgh Mathematical Society 1976

References

REFERENCES

(1) Chatters, A. W. and Ginn, S. M., Localisation in hereditary rings, J. Algebra 22 (1972), 82–88.CrossRefGoogle Scholar
(2) Gordon, R. and Robson, J. C., Krull dimension, Memoirs American Math. Soc. 133 (1973).Google Scholar
(3) Krause, G., On the Krull dimension of left Noetherian, left Matlis rings, Math. Z. 118 (1970), 207–214.CrossRefGoogle Scholar
(4) Rentschler, R. and Gabriel, P., Sur la dimension des anneaux et ensembles ordonnes, C.R. Acad. Sci. Paris 265 (1967), 712–715.Google Scholar
(5) Smith, P. F., On the dimension of group rings, Proc. London Math. Soc. 25 (1972), 288–302; Corrigendum, 27 (1973), 766–768.CrossRefGoogle Scholar
(6) Walker, R., Local rings and normalising sets of elements, Proc. London Math. Soc. 24 (1972), 27–45.CrossRefGoogle Scholar