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Keywords:
$S$-Noetherian ring; $S$-coherent ring; $S$-injective module; $S$-FP-injective module
$S$-Noetherian ring; $S$-coherent ring; $S$-injective module; $S$-FP-injective module
Summary:
Let $R$ be a commutative ring and $S$ a multiplicative subset of $R$. This paper is a sequel to previous works, where we established some module-theoretic characterizations of $S$-Noetherian and $S$-coherent rings. We show that, under certain conditions, a ring $R$ is \hbox {$S$-Noetherian} (or $S$-coherent) if and only if the class of all $S$-injective (or $S$-FP-injective) $R$-modules is (pre)covering. These results provide the $S$-counterpart of Enochs's (or Pinzon, Dai, and Ding's) characterizations for Noetherian (or coherent) rings using the notion of (pre)covering.
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