×

Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings. (English) Zbl 1299.16020

A ring \(R\) is called a near pseudo-valuation ring if every minimal prime ideal of \(R\) is strongly prime. Now let \(\sigma\) be an automorphism of \(R\) and let \(\delta\) be a \(\sigma\)-derivation of \(R\). Then \(R\) is called an almost \(\delta\)-divided ring if every minimal prime ideal of \(R\) is \(\delta\)-divided.
In the present paper the author studies the question when a Noetherian ring \(R\), which is also an algebra over the field of rational numbers, is either a near pseudo-valuation ring or an almost \(\delta\)-divided ring, where \(\delta\) is a \(\sigma\)-derivation of \(R\) with the automorphism \(\sigma\).

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
16D25 Ideals in associative algebras
16P40 Noetherian rings and modules (associative rings and algebras)
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] D.F. Anderson: Comparability of ideals and valuation overrings. Houston J. Math. 5 (1979), 451–463. · Zbl 0407.13001
[2] D.F. Anderson: When the dual of an ideal is a ring. Houston J. Math. 9 (1983), 325–332. · Zbl 0526.13015
[3] A. Badawi: On divided commutative rings. Commun. Algebra 27 (1999), 1465–1474. · Zbl 0923.13001 · doi:10.1080/00927879908826507
[4] A. Badawi: On domains which have prime ideals that are linearly ordered. Commun. Algebra 23 (1995), 4365–4373. · Zbl 0843.13007 · doi:10.1080/00927879508825469
[5] A. Badawi: On {\(\phi\)}-pseudo-valuation Rings. Advances in Commutative Ring Theory (D.E. Dobbs et al., eds.). Proceedings of the 3rd International Conference, Fez, Morocco Lect. Notes Pure Appl. Math. 205, Marcel Dekker, New York, 1999, pp. 101–110.
[6] A. Badawi: On pseudo-almost valuation domains. Commun. Algebra 35 (2007), 1167–1181. · Zbl 1113.13001 · doi:10.1080/00927870601141951
[7] A. Badawi, D.F. Anderson, D. E. Dobbs: Pseudo-valuation Rings. Commutative Ring Theory (P.-J. Cahen et al., eds.). Proceedings of the 2nd International Conference, Fes, Morocco, June 5–10, 1995. Lect. Notes Pure Appl. Math. 185, Marcel Dekker, New York, 1997, pp. 57–67.
[8] A. Badawi, E. Houston: Powerful ideals, strongly primary ideals, almost pseudo-valuation domains, and conducive domains. Commun. Algebra 30 (2002), 1591–1606. · Zbl 1063.13017 · doi:10.1081/AGB-120013202
[9] H.E. Bell, G. Mason: On derivations in near-rings and rings. Math. J. Okayama Univ. 34 (1992), 135–144. · Zbl 0810.16042
[10] V.K. Bhat: A note on completely prime ideals of Ore extensions. Int. J. Algebra Comput. 20 (2010), 457–463. · Zbl 1194.16020 · doi:10.1142/S021819671000573X
[11] V.K. Bhat: On near pseudo valuation rings and their extensions. Int. Electron. J. Algebra (electronic only) 5 (2009), 70–77. · Zbl 1162.13304
[12] V.K. Bhat: Polynomial rings over pseudovaluation rings. Int. J. Math. Math. Sci. 2007 (2007), Article ID 20138, 6 pages. · Zbl 1167.16021
[13] V.K. Bhat, N. Kumari: On Ore extensions over near pseudo valuation rings. Int. J.Math. Game Theory Algebra 20 (2011), 69–77. · Zbl 1316.16019
[14] V.K. Bhat, N. Kumari: Transparency of {\(\sigma\)}(*)-rings and their extensions. Int. J. Algebra 2 (2008), 919–924. · Zbl 1175.16022
[15] K.R. Goodearl, R.B. Warfield Jr.: An Introduction to Noncommutative Noetherian Rings. London Mathematical Society Student Texts 16, Cambridge University Press, Cambridge, 1989. · Zbl 0679.16001
[16] J.R. Hedstrom, E.G. Houston: Pseudo-valuation domains. Pac. J. Math. 75 (1978), 137–147. · Zbl 0368.13002 · doi:10.2140/pjm.1978.75.137
[17] J. Krempa: Some examples of reduced rings. Algebra Colloq. 3 (1996), 289–300. · Zbl 0859.16019
[18] T.K. Kwak: Prime radicals of skew polynomial rings. Int. J. Math. Sci. 2 (2003), 219–227. · Zbl 1071.16024
[19] J.C. McConnell, J.C. Robson: Noncommutative Noetherian Rings. With the cooperation of L.W. Small. Reprinted with corrections from the 1987 original. Graduate Studies in Mathematics 30, American Mathematical Society, Providence, 2001. · Zbl 0980.16019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.