Monomorphic flat envelopes in commutative rings (Q1111623)

\textit{E. Enochs} [Isr. J. Math. 39, 189-209 (1981; Zbl 0464.16019)] defined flat envelopes and proposed the problem of characterizing the rings R with flat envelope for every R-module. Unlike the injective hull, the flat envelope need not be a monomorphism. In the present paper the commutative rings with monomorphic flat envelope for each module are characterized as the coherent rings for which the localization at each maximal ideal is a quasi-Frobenius ring. The quasi- Frobenius rings are precisely the rings over which the flat envelope of each module coincides with its injective hull. In the commutative case, the coherent self-injective rings are characterized by the condition that every finitely presented module has a flat envelope which coincides with its injective hull, or equivalently, those self-injective rings for which the injective hull of each finitely presented module is a finitely generated projective module.