Polycyclic group rings and unique factorisation rings

MacKenzie, Kenneth W.

Glasgow mathematical journal, Tome 36 (1994) no. 2, pp. 135-144

Résumé

The theory of unique factorisation in commutative rings has recently been extended to noncommutative Noetherian rings in several ways. Recall that an element x of a ring R is said to be normalif xR = Rx. We will say that an element p of a ring R is (completely) prime if p is a nonzero normal element of R and pR is a (completely) prime ideal. In [2], a Noetherian unique factorisation domain (or Noetherian UFD) is defined to be a Noetherian domain in which every nonzero prime ideal contains a completely prime element: this concept is generalised in [4], where a Noetherian unique factorisation ring(or Noetherian UFR) is defined as a prime Noetherian ring in which every nonzero prime ideal contains a nonzero prime element; note that it follows from the noncommutative version of the Principal Ideal Theorem that in a Noetherian UFR, if pis a prime element then the height of the prime ideal pR must be equal to 1. Surprisingly many classes of noncommutative Noetherian rings are known to be UFDs or UFRs: see [2] and [4] for details. This theory has recently been extended still further, to cover certain classes of non-Noetherian rings: see [3].

Détail

DOI : 10.1017/S0017089500030676

Bibliographie

[1] 1.Brown, K. A., Height one primes of polycyclic group rings, J. London Math. Soc.(2) 32 (1985), 426–438. Google Scholar

[1'] 1'.Brown, K. A., Corrigendum and addendum. Height one primes of polycyclic group rings, J. London Math. Soc.(2) 38 (1988), 421–22. Google Scholar

[2] 2.Chatters, A. W., Noncommutative unique factorisation domains. Math. Proc. Cambridge Philos. Soc. 95 (1984), 49–54. Google Scholar | DOI

[3] 3.Chatters, A. W., Gilchrist, M. P. and Wilson, D., Unique factorisation rings, Proc. Edinburgh Math. Soc.(2) 35 (1992), 255–269. Google Scholar

[4] 4.Chatters, A. W. and Jordan, D. A., Noncommutative unique factorisation rings, J. London Math. Soc.(2) 33 (1986), 22–32. Google Scholar

[5] 5.Gilchrist, M. P. and Smith, M. K., Noncommutative UFDs are often PIDs, Math. Proc. Cambridge Philos. Soc. 95 (1984), 417–419. Google Scholar

[6] 6.Hajarnavis, C. R. and Lenagan, T. H., Localisation in Asano orders, J. Algebra 21 (1972), 441–449. Google Scholar

[7] 7.Jategaonkar, A. V., Localization in Noetherian rings, London Math. Soc. Lecture Note Series 98 (Cambridge University Press, 1986). Google Scholar

[8] 8.McConnell, J. C. and Robson, J. C., Noncommutative Noetherian rings(Wiley-Interscience, 1987). Google Scholar

[9] 9.Passman, D. S., The algebraic structure of group rings(Wiley-Interscience, 1977), (Krieger, 1985). Google Scholar

[10] 10.Passman, D. S., Infinite crossed products(Academic Press, 1989). Google Scholar

[11] 11.Smith, P. F., Quotient rings of group rings, J. London Math. Soc. (2) 3 (1971), 645–660. Google Scholar