Polynomial Rings over Pseudovaluation Rings

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.