ON NEAR 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 ring R is near pseudo-valuation ring if every minimal prime ideal 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 ring R is almost δ-divided ring if every minimal prime ideal of R is δ-divided. With this we prove the following: Let R be a commutative Noetherian Q-algebra (Q is the field of rational numbers), σ and δ as usual. Then: (1) If R is a near pseudo valuation σ(∗)- ring, then R[x; σ, δ] is a near pseudo valuation ring. (2) If R is an almost δ-divided σ(∗)-ring, then R[x; σ, δ] is an almost divided ring.