When is $F[x,y]$ a unique factorization domain?
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- by Raymond A. Beauregard PDF
- Proc. Amer. Math. Soc. 117 (1993), 67-70 Request permission
Abstract:
Although the commutative polynomial ring $F[x,y]$ is a unique factorization domain (UFD) and the free associative algebra $F\langle x,y\rangle$ is a similarity-UFD when $F$ is a (commutative) field, it is shown that the polynomial ring $F[x,y]$ in two commuting indeterminates is not a UFD in any reasonable sense when $F$ is the skew field of rational quaternions.References
- P. M. Cohn, Unique factorization domains, Amer. Math. Monthly 80 (1973), 1–18. MR 320044, DOI 10.2307/2319253
- P. M. Cohn, Free rings and their relations, 2nd ed., London Mathematical Society Monographs, vol. 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. MR 800091
- John B. Fraleigh, A first course in abstract algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967. MR 0225619
- L. Rédei, Algebra. Vol 1, Pergamon Press, Oxford-New York-Toronto, Ont., 1967. Translated from the Hungarian. MR 0211820
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 117 (1993), 67-70
- MSC: Primary 16U30
- DOI: https://doi.org/10.1090/S0002-9939-1993-1132407-8
- MathSciNet review: 1132407