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Unique factorization in the ring R[x]

Part of: Conditions on elements Rings and algebras arising under various constructions

Published online by Cambridge University Press:  09 April 2009

Raymond A. Beauregard
Raymond A. Beauregard
Affiliation:
University of Rhode IslandKingston, R.I. 02881, USA
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Abstract

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If R is a commutative unique factorization domain (UFD) then so is the ring R[x]. If R is not commutative then no such result is possible. An example is given of a bounded principal right and left ideal domain R, hence a similarity-UFD, for which the polynomial ring R[x] in a central indeterminate x is not a UFD in any reasonable sense. On the other hand, it is shown that if R is an invariant UFD then R[x] is a UFD in an appropriate sense.

MSC classification

Secondary: 16U30: Divisibility, noncommutative UFDs 16S36: Ordinary and skew polynomial rings and semigroup rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 53 , Issue 3 , December 1992 , pp. 287 - 293
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Beauregard, R. A., ‘Right bounded factors in an LCM domain’. Trans. Amer. Math. Soc. 200 (1974), 251–266.CrossRefGoogle Scholar
[2]Beauregard, R. A., ‘An analog of Nagata's Theorem for modular LCM domains’, Canad. J. Math. 29 (1977), 307–314.CrossRefGoogle Scholar
[3]Chatters, A. W. and Jordan, A. A., ‘Noncommutative unique factorisation rings’, J. London Math Soc. (2) 33 (1986), 22–32.Google Scholar
[4]Cohn, P. M., ‘Unique factorization domains’, Amer. Math. Monthly 80 (1973), 1–18.CrossRefGoogle Scholar
[5]Cohn, P. M., Free rings and their relations (2nd ed.) (Academic Press, London, 1985).Google Scholar
[6]Redei, L, Algebra (Vol 1) (Pergamon Press, London, 1967).Google Scholar