On semi-pseudo-valuation rings and their extensions

Abstract

Recall that a commutative ring R is said to be a pseudo-valuation ring (PVR) if every prime ideal of R is strongly prime. We say that a commutative Noetherian ring R is semi-pseudo-valuation ring if assassinator of every right ideal (which is uniform as a right R-module) is a strongly prime ideal. We also recall that a prime ideal P of a ring R is said to be divided if it is comparable (under inclusion) to every ideal of R. A ring R is called a divided ring if every prime ideal of R is divided. Let R be a commutative ring, σ an automorphism of R and δ a σ-derivation of R. We say that a prime ideal P of R is δ-divided if it is comparable (under inclusion) to every σ-stable and δ-invariant ideal I of R. A ring R is called a δ-divided ring if every prime ideal of R is δ-divided. We say that a Noetherian ring R is semi-δ-divided ring if assassinator of every right ideal (which is uniform as a right R-module) is δ-divided. Let R be a ring and σ an endomorphism of R. Recall that R is said to be a σ(*)-ring if aσ(a) ∈ P(R) implies that a ∈ P(R), where P(R) is the prime radical of R. With this we prove the following

Let R be a semiprime commutative Noetherian ℚ-algebra, σ an automorphism of R such that R is a σ(*)-ring and δ a σ-derivation of R. Then:

• (1) If any U ∈ S.Spec(R) with σ(U) = U and δ(U) ⊆ U implies that O(U) ∈ S.Spec(O(R)), then R is a semi-pseudo-valuation ring implies that R[x; σ, δ] is a semi-pseudo-valuation ring.

• (2) If R is a semi-δ-divided ring, then O(R) is a Noetherian semi-δ-divided ring.