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Unique factorisation rings

Published online by Cambridge University Press:  20 January 2009

A. W. Chatters ,
M. P. Gilchrist  and
D. Wilson
A. W. Chatters
Affiliation:
School of MathematicsUniversity of BristolUniversity WalkBristol BS8 1TN
M. P. Gilchrist
Affiliation:
Science and Medical DepartmentOxford University PressWalton StreetOxford OX2 6DP
D. Wilson
Affiliation:
Mathematical InstituteUniversity of WarwickCoventry CV4 7AL
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Abstract

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Let R be a ring. An element p of R is a prime element if pR = Rp is a prime ideal of R. A prime ring R is said to be a Unique Factorisation Ring if every non-zero prime ideal contains a prime element. This paper develops the basic theory of U.F.R.s. We show that every polynomial extension in central indeterminates of a U.F.R. is a U.F.R. We consider in more detail the case when a U.F.R. is either Noetherian or satisfies a polynomial identity. In particular we show that such a ring R is a maximal order, that every height-1 prime ideal of R has a classical localisation in which every two-sided ideal is principal, and that R is the intersection of a left and right Noetherian ring and a simple ring.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 35 , Issue 2 , June 1992 , pp. 255 - 269
Copyright
Copyright © Edinburgh Mathematical Society 1992

References

REFERENCES

1.Brown, K. A., Height one primes of polycyclic group rings, J. London Math. Soc. (2) 32 (1985), 426–438.CrossRefGoogle Scholar
2.Cauchon, G., Anneaux semipremiers Noetheriens, a identites polynomials, Bull. Soc. Math. France 104 (1976), 99–111.CrossRefGoogle Scholar
3.Chamarie, M., Localisations dans les ordres maximaux, Comm. Algebra 2 (1974), 279–293.Google Scholar
4.Chatters, A. W. and Hajarnavis, C. R., Rings with chain conditions (Pitman, London, 1980).Google Scholar
5.Chatters, A. W., Non-commutative unique factorisation domains, Math. Proc. Cambridge Philos. Soc. 95 (1984), 49–54.CrossRefGoogle Scholar
6.Chatters, A. W. and Jordan, D. A., Non-commutative unique factorisation rings, J. London Math. Soc. (2) 33 (1986), 22–32.Google Scholar
7.Chatters, A. W. and Clark, J., Group-rings which are unique factorisation rings, Comm. Algebra, to appear.Google Scholar
8.Formanek, E. and Jategaonkar, A. V., Subrings of Noetherian rings, Proc. Amer. Math. Soc. 46(1974), 181–186.CrossRefGoogle Scholar
9.Gilchrist, M. P. and Smith, M. K., Non-commutative UFD's are often PID's, Math. Proc. Cambridge Philos. Soc. 95 (1984), 417–419.Google Scholar
10.Gilmer, R., Multiplicative ideal theory (Marcel Dekker Inc., 1982).Google Scholar
11.Jordan, D. A., Unique factorisation of normal elements in non-commutative rings, Glasgow Math. J. 31 (1989), 103–113.Google Scholar
12.Kaplansky, I., Commutative rings, revised edition (U. of Chicago Press, 1974).Google Scholar
13.Le Bruyn, L., Trace rings of generic matrix rings are unique factorisation domains, Glasgow Math. J. 28(1986), 11–13.Google Scholar
14.Maury, G. and Raynaud, J., Ordres maximaux au sens de K. Asano (Springer Lecture Notes in Mathematics Vol. 808).Google Scholar
15.Mcconnell, J. C. and Robson, J. C., Non-commutative Noetherian rings (Wiley, 1987).Google Scholar
16.Rowen, L. S., Polynomial identities in ring theory (Academic Press, 1980).Google Scholar
17.Stafford, J. T., Noetherian full quotient rings, Proc. London Math. Soc. (3) 45 (1982), 385–404.CrossRefGoogle Scholar
18.Stafford, J. T., Modules over prime Krull rings, J. Algebra 95 (1985), 332–342.CrossRefGoogle Scholar