Filtered Ring Derived from Discrete Valuation Ring and Its Properties


In this paper we show that if R is a discrete valuation ring, then R is a filtered ring. We prove some properties and relation when R is a discrete valuation ring.

Share and Cite:

Shoa, M. and Hosseini, M. (2014) Filtered Ring Derived from Discrete Valuation Ring and Its Properties. Advances in Pure Mathematics, 4, 71-75. doi: 10.4236/apm.2014.43011.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Atiyah, M.F. and Macdonald, L.G. (1969) Introduction to Commutative Algebra. Addison-Wesley Publishing Company.
[2] Bourbaki, N. (1972) Commutative Algebra. Originally Published as Elements de Mathematique, Algebra Commutative 1964, 1965, 1968, 1969 by Hermann, Paris.
[3] Gopalakrishnan, N.S. (1983) Commutative Algebra. Oxonian Press, PVT, LTD, New Delhi.
[4] Puninskia, G., Puninskayab, V. and Toffalorib, C. (2007) Decidability of the Theory of Modules over Commutative Valuation Domains. Annals of Pure and Applied Logic, 145, 258-275.
[5] Cohen, F.R. and Heap, A. (2011) Alexandra Pettet on the Andreadakis Johnson Filtration of the Automorphism Group of a Free Group. Journal of Algebra, 329, 72-91.
[6] Levy, R., Loustauna, P. and Shapiro, J. (1991) The Prime Spectrum of an Infinite Product of Copies of Z. Fundamenta Mathematicae, 138, 155-164.
[7] Nishida, K. (2005) On the Depth of the Associated Graded Ring of a Filtration. Journal of Algebra, 285, 182-195.
[8] Olberding, B., Saydam, S. and SHapiro, J. (2005) Complitions, Valuations and Ultrapowers of Noetherian Domain. Journal of Pure and Applied Algebra, 197, 213-237.
[9] Rush, D.E. (2007) Rees Valuations and Asymptotic Primes of Rational Powers in Noetherian Rings and Lattices. Journal of Algebra, 308, 295-320.

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.