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) cM = I for some nonzero c E 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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