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Almost Krull rings

Published online by Cambridge University Press:  09 April 2009

E. Jespers  and
P. Wauters
E. Jespers
Affiliation:
Department of Mathematics, University of Cape Town, 7700 Rondebosch, South Africa
P. Wauters
Affiliation:
Katholieke Universiteit Leuven, Leuven, Belgium
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Abstract

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The notion of an almost Krull domain is extended to rings satisfying a polynomial identity. Some general structural results are obtained. We also prove that skew polynomial rings R [ X, σ] remain almost Krull if R is an almost Krull ring. Finally, we study when semigroup ring R[S] are almost Krull rings, in the case when the group of quotients of S has the ascending chain condition on cyclic subgroups. An example is included to show that the general (semi-) group ring case is much more difficult to deal with.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 2 , October 1986 , pp. 275 - 285
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Amitsur, S. A. and Small, L. W., ‘Prime ideals in PI-rings’, J. Algebra 62 (1980), 358–383.CrossRefGoogle Scholar
[2]Bourbaki, N., Commutative algebra, Vol. 8 (Hermann, Paris, 1972).Google Scholar
[3]Cauchon, G., Les T-anneaux et les anneaux à identités polynomiales Noétheriens (Thése, Université de Paris XI 1977).Google Scholar
[4]Chamaire, M., Anneaux de Krull non commutatifs (Thése, Université Claude-Bernard-Lyon I 1981).CrossRefGoogle Scholar
[5]Chamarie, M., ‘Anneaux de Krull non commutatifs’, J. Algebra 72 (1981), 210–222.Google Scholar
[6]Chouinard, L. G. II, ‘Krull semigroups and divisor class groups’, Canad. J. Math. 23 (1981), 1459–1468.CrossRefGoogle Scholar
[7]Fossum, R. M., ‘Maximal orders over Krull domains’, J. Algebra 10 (1968), 321–332.CrossRefGoogle Scholar
[8]Fossum, R. M., The divisor class group of a Krull domain (Springer Verlag, Berlin, 1973).CrossRefGoogle Scholar
[9]Gilmer, R., Multiplicative ideal theory (Marcel Dekker, New York,1972).Google Scholar
[10]Gilmer, R. and Parker, T., ‘Divisibility properties in semigroup rings’, Michigan Math. J. 21 (1974), 65–86.CrossRefGoogle Scholar
[11]Larsen, M. D. and McCarthy, P. J., Multiplicative theory of ideals (Academic Press, New York, 1971).Google Scholar
[12]Matsuda, R., ‘On algebraic properties of infinite group rings’, Bull. Fac. Sci. Ibaraki Univ. Series A 7 (1975), 29–37.CrossRefGoogle Scholar
[13]Maury, G. and Raynaud, J., Orders maximaux au sens de K. Asano (Lecture Notes in Math. Vol. 808, Springer Verlag, Berlin, 1980).Google Scholar
[14]Passman, D. S., ‘It's essentially Maschke's theorem’, Rocky Mountain J. Math. 13 (1983), 37–54.CrossRefGoogle Scholar
[15]Robson, J. C. and Small, L. W., ‘Hereditary prime P. I. rings are classical hereditary orders’, J. London Math. Soc. 8 (1974), 499–503.CrossRefGoogle Scholar
[16]Rowen, L. H., Polynomial identities in ring theory (Academic Press New York, 1977).Google Scholar
[17]Schelter, W., ‘Integral extensions of rings satisfying a polynomial identity’, J. Algebra 40 (1976), 245–257.Google Scholar
[18]Wauters, P., ‘On Ω-Krull rings in the graded sense’, Bull. Soc. Math. Belg. 35 (1983), 1–26.Google Scholar
[19]Wauters, P., ‘On some subsemigroups of noncommutative Krull rings’, Comm. Algebra 12 (1984), 1751–1765.Google Scholar