On S-injective and S-FP-injective (pre)covers

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 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.