Valuations and rings of quotients

Brown, David E.

doi:10.1090/S0002-9939-1974-0347792-2

Valuations and rings of quotients
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by David E. Brown PDF

Proc. Amer. Math. Soc. 43 (1974), 277-282 Request permission

Abstract:

Valuations on a commutative ring, as defined by Manis, are considered in the special case where the domain of the valuation mapping is a ring of quotients of a given ring $R$. We consider relations between valuation mappings on various rings of quotients of a given ring. It is also shown that if $K$ is any von Neumann regular ring of quotients of $R$, then any pair of nonassociates of $R$ can be separated by valuations on $K$ if and only if these elements are nonassociates in the integral closure of $R$ in $K$.

References

David E. Brown and Max D. Larsen, Separation of nonassociates by valuations, Proc. Amer. Math. Soc. 31 (1972), 326–332. MR 289506, DOI 10.1090/S0002-9939-1972-0289506-9
N. J. Fine, L. Gillman, and J. Lambek, Rings of quotients of rings of functions, McGill University Press, Montreal, Que., 1966. MR 0200747
Malcolm Griffin, Prüfer rings with zero divisors, J. Reine Angew. Math. 239(240) (1969), 55–67. MR 255527, DOI 10.1515/crll.1969.239-240.55
Joachim Lambek, Lectures on rings and modules, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. With an appendix by Ian G. Connell. MR 0206032
Merle E. Manis, Valuations on a commutative ring, Proc. Amer. Math. Soc. 20 (1969), 193–198. MR 233813, DOI 10.1090/S0002-9939-1969-0233813-2
A. C. Mewborn, Some conditions on commutative semiprime rings, J. Algebra 13 (1969), 422–431. MR 276209, DOI 10.1016/0021-8693(69)90084-2