Let R be a domain with quotient ®eld Q. R is divisorial if R X R X I I for every nonzero fractional ideal I of R. We prove that a local domain R, not a ®eld, is divisorial if and only if QaR has simple essential socle and RarR is AB-5* for every nonzero r e R. We give examples of non-divisorial and of non-®nitely divisorial local domains such that QaR 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 X A X 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 ®nitely 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 Adivisorial if and only if QaA has simple essential socle and RarR is AB-5* for every nonzero r e R.