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ON NEAR PSEUDO-VALUATION RINGS AND THEIR EXTENSIONS

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

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.

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: vijaykumarbhat2000@yahoo.com
Year 2009, Volume: 5 Issue: 5, 70 - 77, 01.06.2009

Abstract

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: vijaykumarbhat2000@yahoo.com
There are 13 citations in total.

Details

Other ID JA43ZG96JZ
Journal Section Articles
Authors

V. K. Bhat This is me

Publication Date June 1, 2009
Published in Issue Year 2009 Volume: 5 Issue: 5

Cite

APA Bhat, V. K. (2009). ON NEAR PSEUDO-VALUATION RINGS AND THEIR EXTENSIONS. International Electronic Journal of Algebra, 5(5), 70-77.
AMA Bhat VK. ON NEAR PSEUDO-VALUATION RINGS AND THEIR EXTENSIONS. IEJA. June 2009;5(5):70-77.
Chicago Bhat, V. K. “ON NEAR PSEUDO-VALUATION RINGS AND THEIR EXTENSIONS”. International Electronic Journal of Algebra 5, no. 5 (June 2009): 70-77.
EndNote Bhat VK (June 1, 2009) ON NEAR PSEUDO-VALUATION RINGS AND THEIR EXTENSIONS. International Electronic Journal of Algebra 5 5 70–77.
IEEE V. K. Bhat, “ON NEAR PSEUDO-VALUATION RINGS AND THEIR EXTENSIONS”, IEJA, vol. 5, no. 5, pp. 70–77, 2009.
ISNAD Bhat, V. K. “ON NEAR PSEUDO-VALUATION RINGS AND THEIR EXTENSIONS”. International Electronic Journal of Algebra 5/5 (June 2009), 70-77.
JAMA Bhat VK. ON NEAR PSEUDO-VALUATION RINGS AND THEIR EXTENSIONS. IEJA. 2009;5:70–77.
MLA Bhat, V. K. “ON NEAR PSEUDO-VALUATION RINGS AND THEIR EXTENSIONS”. International Electronic Journal of Algebra, vol. 5, no. 5, 2009, pp. 70-77.
Vancouver Bhat VK. ON NEAR PSEUDO-VALUATION RINGS AND THEIR EXTENSIONS. IEJA. 2009;5(5):70-7.