A Characterization of Valuation Domains via m-Canonical Ideals

Abstract

A nonzero ideal I of an intergral domain R is said to be an m-canonical ideal of R if (I : (I : J)) = J for every nonzero ideal J of R. In this paper, we show that if a quasi-local integral domain (R, M) admits a proper m-canonical ideal I of R, then the following statements are equivalent:

1.	R is a valuation domain.
2.	I is a divided m-canonical ideal of R.
3.	c M = I for some nonzero c ∈ R.
4.	(I : M) is a principal ideal of R.
5.	(I : M) is an invertible ideal of R.
6.	R is an integrally closed domain and (I : M) is a finitely generated of R.
7.	(M : M) = R and (I : M) is a finitely generated of R.
8.	If J = (I : M), then J is a finitely generated of R and (J : J) = R.

Among the many results in this paper, we show that an integral domain R is a valuation domain if and only if R admits a divided proper m-canonical ideal, iff R is a root closed domain which admits a strongly primary proper m-canonical ideal, also we show that an integral domain R is a one-dimensional valuation domain if and only if R is a completely integrally closed domain which admits a powerful proper m-canonical ideal of R.

Key Words:

• m-Canonical ideals
• Valuation domains
• Divided ideals
• Prüfer domain

Mathematics Subject Classification:

• 13A15
• 13F05
• 13F30

Acknowledgment

The author would like to thank the referee for his(her) helpful suggestions which helped in improving the quality of this paper.

Notes

^#Communicated by I. Swanson.