Holder valuation and holder rigidity for right ring of fractions

Abstract

The purpose of this article is to introduce the notion of (C1,C2)-Holder Krull valuation on right ring of fractions ( with respect to right denominator set S in a ring R). It is proved that if R is a ring satisfying in Holder rigidity condition, and S a right permutable set of regular elements in R, then the right ring of fractions R′= Q _Â(R)  with respect to S satisfies in H¨older  rigidity  condition. This results provide an extension of the Garsia theorem (see [2]) for right ring of fractions.

Keywords

• Valuation,Holder valuation,Holder equivalent

References

1. O. Endler, Valuation Theory, New York, Springer-Verlag, (1972).
2. E. Munoz Garcia, Holder absolute values are equivalent to classicalones, Proc. Amer. Math. Soc, 127(7), 1967-1971, (1999).
3. J. A. Huckaba, Commutative Rings with Zero Divisors, New York, Marcel Dekker, Inc (1988).
4. M. Knebusch, D. Zhang, Manis Valuations and Prufer Extensions, I. A new chapter in commutative algebra. Lecture Notes in Math. 1791. Berlin, Springer-Verlage, (2002).
5. T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, (1991).
6. M. Manis, Valuations on a commutative ring, Proc. Amer. Math. Soc, 20, 193-198, (1969).
7. P. Ribenboim, The theorie des Valuations, Montreal, Less presses de I Universite de Montreal (1964).
8. O. F. G. Schilling, The theory of Valuations, New York, Amer. Math. Soc, Thomas, J. J, (1978). Set theory, New York, Academic press, Inc (1952).