Flat envelopes in commutative rings (Q1117274)

The authors take up the notion of a flat preenvelope and a flat envelope of a module over an associative ring R, first introduced by \textit{E. E. Enochs} [Isr. J. Math. 39, 189-209 (1981; Zbl 0464.16019)], and persue some of his results. Particularly, the relation between R being coherent and the existence of envelopes and preenvelopes is discussed. If R is commutative and its localization for each prime ideal has a finite weak global dimension, then every R-module has a flat envelope if and only if R is coherent with a weak global dimension of \(at\quad most\quad 2\).