International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 20138, 6 pages
doi:10.1155/2007/20138
Research Article

Polynomial Rings over Pseudovaluation Rings

V. K. Bhat

School of Applied Physics and Mathematics, Shri Mata Vaishno Devi University, P/o Kakryal, Katra, 182301, India

Received 5 March 2007; Accepted 1 August 2007

Academic Editor: Howard E. Bell

Copyright © 2007 V. K. Bhat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let R be a ring. Let σ be an automorphism of R. We define a σ-divided ring and prove the following. (1) Let R be a commutative pseudovaluation ring such that x∉P for any P∈Spec(R[x,σ]) . Then R[x,σ] is also a pseudovaluation ring. (2) Let R be a σ-divided ring such that x∉P for any P∈Spec(R[x,σ]). Then R[x,σ] is also a σ-divided ring. Let now R be a commutative Noetherian Q-algebra (Q is the field of rational numbers). Let δ be a derivation of R. Then we prove the following. (1) Let R be a commutative pseudovaluation ring. Then R[x,δ] is also a pseudovaluation ring. (2) Let R be a divided ring. Then R[x,δ] is also a divided ring.