ON NEAR PSEUDO-VALUATION RINGS AND THEIR EXTENSIONS

Year 2009, Volume: 5 Issue: 5, 70 - 77, 01.06.2009

V. K. Bhat

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.

Keywords

Ore extension, automorphism, derivation, divided prime, almost divided prime, pseudo-valuation ring, near pseudo-valuation ring

References

• D. F. Anderson, Comparability of ideals and valuation rings, Houston J. Math., (1979), 451-463.
• D. F. Anderson, When the dual of an ideal is a ring, Houston J. Math., 9 (1983), 325-332.
• A. Badawi, D. F. Anderson, and D. E. Dobbs, Pseudo-valuation rings, Lecture Notes Pure Appl. Math., 185 (1997), 57-67, Marcel Dekker, New York.
• A. Badawi, On domains which have prime ideals that are linearly ordered, Comm. Algebra, 23 (1995), 4365-4373.
• A. Badawi, On φ-pseudo-valuation rings, Lecture Notes Pure Appl. Math., 205 (1999), 101-110, Marcel Dekker, New York.
• A. Badawi, On divided commutative rings, Comm. Algebra, 27 (1999), 1465
• A. Badawi, On pseudo-almost valuation rings, Comm. Algebra, 35 (2007), 1181.
• H. E. Bell and G. Mason, On derivations in near-rings and rings, Math. J. Okayama Univ., 34 (1992), 135-144.
• V. K. Bhat, Polynomial rings over pseudovaluation rings, Int. J. Math. Math. Sci., (2007), Art. ID 20138.
• K. R. Goodearl and R. B. Warfield Jr, An introduction to non-commutative Noetherian rings, Cambridge University Press, 1989.
• J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains, Pacific J. Math., 4 (1978), 551-567.
• T. K. Kwak, Prime radicals of skew-polynomial rings, Int. J. Math. Sci., 2(2) (2003), 219-227. V. K. Bhat
• School of Mathematics SMVD University Kakryal, Katra, J and K , India e-mail: [e-mail]