NOTE ON PSEUDO-VALUATION DOMAINS WHICH ARE NOT VALUATION DOMAINS
In this article, we discuss the n-root closedness, root closedness, seminormality, S-root closedness, S-closedness, F-closedess of PVDs. A valuation domain, being integrally closed, is obviously root closed. So our interest of study is for a class of non-valuation PVDs. Let R ⊂ B be a domain extension such that R is a PVD and the common ideal P of R and B is a prime ideal in R. If R is n-root closed (respectively root closed, seminormal, S-root closed, S-closed, F-closed) in B, then R/P is PVD, which is n-root closed (respectively root closed, seminormal, S-root closed, S-closed, F-closed) in B/P. Further we study the relationship of atomic PVDs to atomic PVDs, SHFDs, LHFDs and BVDs. We also discuss a relative ascent and descent in general and particularly for the antimatter property of PVDs.
Keywords:
PVD, atomic domain, F + M construction, root-closed, factor ring condition ∗, antimatter domain,
___
- D. D. Anderson, D. F. Anderson and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra, 69 (1990), 1-19.
- D. D. Anderson, D. F. Anderson and M. Zafrullah, Factorization in integral domains, J. Algebra, 152 (1992), 78-93.
- D. F. Anderson, Root closure in integral domains, J. Algebra, 79 (1982), 51-59.
- D. F. Anderson and S. T. Chapman, Overrings of half-factorial domains II, Comm. Algebra, 23(11) (1995), 3961-3976.
- D. F. Anderson and D. E. Dobbs, Pairs of rings with the same prime ideals, Canad. J. Math, 32 (1980), 362-384.
- D. F. Anderson and D. E. Dobbs, Root Closure in Integral Domains III, Canad. Math. Bull., 41(1) (1998), 3-9.
- D. F. Anderson, D. E. Dobbs and J. A. Huckaba, On seminormal overrings, Comm. Algebra, 10 (1982), 1421-1448.
- D. F. Anderson, D. E. Dobbs and M. Roitman, Root closure in commutative rings, Ann. Sci. Univ. Clermont-Ferrand II Math., 26 (1990), 1-11.
- A. Badawi, Remarks on pseudo-valuation rings, Comm. Algebra, 28(5) (2000), 2358.
- A. Badawi, D. F. Anderson and D. E. Dobbs, Pseudo-valuation Rings, Proc. Second International conference on comm. rings, Lecture notes in Pure and applied Maths., 185 (1996), 57-67.
- E. Basttida and R.Gilmer, Overrings and divisorial ideals of rings of the form D + M , Michigan Math. J., 20 (1973), 79-95.
- P. M. Cohn, Bezeout rings and their subrings, Proc. Camb. Phil. Soc., 64 (1968), 251-264.
- J. Coykendall, D. E. Dobbs and B. Mullins, On integral domains with no atoms, Comm. Algebra, 27 (1999), 5813-5831.
- T. Dumitrescu, T. Shah and M. Zafrullah, Domains whose overrings satisfy ACCP, Comm. Algebra, 28(9) (2000), 4403-4409.
- J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains, Pacific J. Math., 75(1) (1978), 137-147.
- J. Maney, Boundary valuation domains, J. Algebra, 273 (2004), 373-383.
- N. Radu, S. O. Ibrahim Al-Salihi and T. Shah, Ascend and descend of factor- ization properties, Rev. Roumaine Math. Pures Appl., 45(4) (2000), 659-669.
- T. Shah, Relative ascent and descent in a domain extension, Int. Electron. J. Algebra, 7 (2010), 34-46.
- J. J. Watkins, Root and integral closure for R[[X]], J. Algebra, 75 (1982), 58.
- M. Zafrullah, On a property of pre-Schreier domains, Comm. Algebra, 15(9) (1987), 1895-1920.
- A. Zaks, Half-factorial domains, Bull. Amer. Math. Soc., 82(5) (1976), 721-723. Tariq Shah
- Department of Mathematics Quaid-I-Azam University Islamabad, Pakistan e-mail: stshah@gmail.com Waheed Ahmad Khan Department of Mathematics and Statistics Caledonian College of Engineering P O Box 2322, Seeb 111, Sultanate of Oman e-mail: sirwak2003@yahoo.com
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI