VALUATION, DISCRETE VALUATION AND DEDEKIND MODULES

Year 2010, Volume: 8 Issue: 8, 18 - 29, 01.12.2010

J. Moghaderi R. Nekooei

Abstract

The purpose of this paper is to introduce valuation and discrete valuation modules over an integral domain. Some basic results and characterizations are obtained and these results are used to characterizeDedekind multiplication modules with discrete multiplication valuation modules.

Keywords

multiplication module, Dedekind module, prime submodule, valuation, discrete valuation, fractional submodule

References

• Z. Abd El-bast and P.F. Smith, Multiplication modules, Comm. Algebra, 16(4) (1988), 755–779.
• M. Alkan, B. Sarac and Y. Tiras, Dedekind modules, Comm. Algebra, (5)(2005), 1617–1626.
• M. Alkan and Y. Tiras, On Invertible and Dense Submodules, Comm. Algebra, (10)(2004), 3911–3919.
• R. Ameri, On The Prime Submodules of Multiplication Modules, IJMMS, (2003), 1715–1724.
• M.F. Atiyah and I.G.MacDonald, Introduction to Commutative Algebra, Addison-Wesley,(1969).
• N. Bourbaki, Commutative Algebra, Addison-Wesley,(1972).
• S. Hedayat and R. Nekooei, Characterization of prime submodules of a finitely generated free module over a PID, Houston J. Math., 31(1)(2005), 75–85.
• S. Hedayat and R. Nekooei, Prime and radical submodules of free modules over a PID, Houston J. Math., 32(2)(2006), 355–367.
• S.Hedayat and R. Nekooei, Primary Decomposition of submodules of a finitely generated module over a PID, Houston J. Math., 32(2)(2006), 369–377.
• S. Karimzadeh and R. Nekooei, On Dimension of Modules, Turkish J. Math, (2007), 95–109.
• S. Karimzadeh and R. Nekooei, Some remarks on Dedekind modules, Accepted in Algebra Colloquium. M.D. Larsen and P.J. McCarthy, Multiplicative theory of ideals, Academic Press, London,(1971).
• A.G. Naoum and F.H. Al-Alwan, Dedekind modules, Comm. Algebra, (2)(1996), 397–412.
• R. Nekooei, On finitely generated multiplication modules, Czechoslovak Math. J., 55(130)(2005), 503–510.
• B.Sarac, P. F. Smith and Y. Tiras, On Dedekind module, Comm. Algebra, 35 (2007) 1532–1538.
• P.F. Smith, Some remarks on multiplication modules, Arch. Math. 50(1988), –235. J. Moghaderi
• Department of Mathematics Hormozgan University Iran e-mail: [e-mail] R. Nekooei Department of Mathematics Shahid Bahonar University of Kerman Iran e-mail: [e-mail]