Abstract
A Matlis domain is a commutative domain R whose ring of quotients Q has projective dimension one. The properties of these domains have been extensively studied in the literature. In [18], Kaplansky proved that the quotient field of a local domain has projective dimension one if and only if it is countably generated. Hamsher, in his beautiful paper [16], showed that if R is a Matlis domain any countably generated submodule of Q/R can be embedded in a countably generated direct summand of Q/R. This result generalizes Kaplansky’s theorem because if R is local then Q/R is indecomposable. Hamsher’s results were completed by Lee in [21] proving that Q has projective dimension at most one if and only if Q/R decomposes into a direct sum of countably generated modules. Fuchs and Salce finally extended these results to arbitrary localizations of commutative domains, see
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