Divisorial domains

Let R be a domain with quotient field Q. R is divisorial if R : (R : I) = I for every nonzero fractional ideal I of R. We prove that a local domain R, not a field, is divisorial if and only if Q/R has simple essential socle and R/rR is AB-5* for every nonzero r ∈ R. We give examples of non-divisorial and of non-finitely divisorial local domains such that Q/R has simple essential socle. If A is any R-submodule of Q with endomorphism ring R, we say that R is A-divisorial if A : (A : X) = X for every nonzero submodule X of A. We prove that if a local noetherian domain R is A-divisorial for some A, then R is one-dimensional and A is finitely generated, i.e. A is isomorphic to a canonical ideal of R. If A is a fractional ideal of R we generalize the characterization of divisorial domains, namely we prove that R is A-divisorial if and only if Q/A has simple essential socle and R/rR is AB-5* for every nonzero r ∈ R.