On generalized divisorial semistar operations on integral domains

Abstract

Throughout this paper the letter D denotes an integral domain with quotient field K. We shall denote the set of all nonzero D-submodules of K by K(D) and we shall call each element of K(D) a Kaplansky fractional ideal (for short, K-fractional ideal ) of D as in [O3]. Let F(D) be the set of all nonzero fractional ideals of D, that is, all elements E ∈ K(D) such that there exists a nonzero element d ∈ D with dE ⊆ D. The set of finitely generated K-fractional ideals of D is denoted by f(D). It is evident that f(D) ⊆ F(D) ⊆ K(D). An ideal of D means an integral ideal of D and the set of all nonzero integral ideals of D is denoted by I(D). If D is a quasi-local domain with maximal ideal M , then we say that (D,M) is a quasi-local domain. In [HHP], a nonzero ideal I of D is called an m-canonical ideal of D if I : (I : J) = J for each nonzero ideal J of D. In [HHP, Proposition 6.2] it was shown that if (D,M) is an integrally closed qausi-local domain, then M is an m-canonical ideal of D if and only if D is a valuation domain. In [BHLP, Proposition 4.1], it was proved that the integrally closed hypothesis in the above result can be eliminated, that is, if (D,M) is a quasi-local domain, then D is a valuation domain if and only if M is an m-canonical ideal of D. Recently, in [B2, Corollary 2.15], it was proved that if a quasi-local integral domain (D,M) admits a proper m-canonical ideal I of D, then the following statements are equivalent: (1) D is a valuation domain. (2) I is a divided m-canonical ideal of D. (3) cM = I for some nonzero element c ∈ D. (4) I : M is a principal ideal of D. (5) I : M is an invertible ideal of D. (6) D is an integrally closed domain and I : M is a finitely generated ideal of D. (7) M : M = D and I : M is a finitely generated ideal of D. (8) If J = I : M , then J is a finitely generated ideal of D and J : J = D. Let I be a nonzero ideal of D such that I : I = D. Then in [HHP, Proposition 3.2], it was proved that the map J 7−→ I : (I : J) of F(D) into F(D) is a star operation