ON NEAR PSEUDO-VALUATION RINGS AND THEIR EXTENSIONS

Authors : V K BHAT

Pages : 70-77

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Publication Date : 2009-06-01

Article Type : Research Paper

Abstract :Recall that a commutative ring R is said to be a pseudo-valuation ring insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(Q is the field of rational numbers);, σ and δ as usual. Then: insert ignore into journalissuearticles values(1); If R is a near pseudo valuation σinsert ignore into journalissuearticles values(∗);- ring, then R[x; σ, δ] is a near pseudo valuation ring. insert ignore into journalissuearticles values(2); If R is an almost δ-divided σinsert ignore into journalissuearticles values(∗);-ring, then R[x; σ, δ] is an almost divided ring.
Keywords : Ore extension, automorphism, derivation, divided prime, almost divided prime, pseudo valuation ring, near pseudo valuation ring