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Keywords:
$S$-coherent ring; $S$-flat module; $S$-flat preenvelope
$S$-coherent ring; $S$-flat module; $S$-flat preenvelope
Summary:
We investigate the notion of $S$-flat preenvelopes of modules. In particular, we give an example that a ring $R$ being coherent does not imply that every $R$-module has an $S$-flat preenvelope, giving a negative answer to the question proposed by D. Bennis and A. Bouziri (2025). Besides, we also show that a ring $R_S$ being coherent also does not imply that $R$ is an $S$-coherent ring in general.
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