No Access

2-Prime ideals and their applications

Charef Beddani

Department of Mathematics, Taibah University, Madinah, Kingdom of Saudi Arabia

E-mail Address: cabeddani@taibahu.edu.sa

Search for more papers by this author

and

Wahiba Messirdi

Department of Mathematics, Taibah University, Madinah, Kingdom of Saudi Arabia

E-mail Address: messirdi.hiba@gmail.com

Search for more papers by this author

https://doi.org/10.1142/S0219498816500511Cited by:15 (Source: Crossref)

Abstract

This paper introduces the notion of $2$ $2$ -prime ideals, and uses it to present certain characterization of valuation rings. Precisely, we will prove that an integral domain $R$ $R$ is a valuation ring if and only if every ideal of $R$ $R$ is $2$ $2$ -prime. On the other hand, we will prove that the normalization $¯ ¯¯ ¯ R$ $\bar{R}$ of $R$ $R$ is a valuation ring if and only if the intersection of integrally closed 2-prime ideals of $R$ $R$ is a 2-prime ideal. At the end of this paper, we will give a generalization of some results of Gilmer and Heinzer by studying the properties of domains in which every primary ideal is an integrally closed 2-prime ideal.

Communicated by S. Ishii

Keywords:

AMSC: 13F30, 13B22, 13G05

References

1. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra (Addison-Wesley, 1969). Google Scholar
2. C. H. Brase, A note on intersections of valuation ideals, Proc. Amer. Math. Soc. 38(1) (1973) 37–39. Crossref, Web of Science, Google Scholar
3. R. Gilmer and W. Heinzer, Primary ideals and valuation ideals II, Trans. Amer. Math. Soc. 131(1) (1968) 149–162. Crossref, Web of Science, Google Scholar
4. R. Gilmer and J. Ohm, Primary ideals and valuation ideals, Trans. Amer. Math. Soc. 114 (1965) 40–52. Crossref, Web of Science, Google Scholar
5. M. Nagata, Local Rings, Interscience Tracts in Pure and Applied Mathematics (Interscience Publishers, 1962). Google Scholar
6. I. Swanson and C. Huneke, Integral Closure of Ideals, Rings, and Modules, London Mathematical Society Lecture Note Series, Vol. 336 (Cambridge University Press, 2006). Google Scholar
7. O. Zariski and P. Samuel, Commutative Algebra, Vol. II (Van Nostrand, New York, 1960). Crossref, Google Scholar