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.
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