Some characterizations of generalized Noetherian rings induced by a multiplicatively closed set

Let [Formula: see text] be a multiplicatively closed set of a commutative ring [Formula: see text]. In this paper, [Formula: see text]-Noetherian rings, i.e. the rings satisfying that every [Formula: see text]-ideal is finitely generated, are studied. In detail, we provide the (Generalized) Principal Ideal Theorem, Hilbert Basis Theorem, Krull–Akizuki Theorem, Special Chinese Remainder Theorem, and Krull Intersection Theorem for [Formula: see text]-Noetherian rings. Then [Formula: see text]-injective modules are discussed. An [Formula: see text]-module [Formula: see text] is called [Formula: see text]-injective if for any [Formula: see text]-ideal [Formula: see text] of [Formula: see text], every homomorphism [Formula: see text] can be extended to [Formula: see text]. The Baer’s Criterion and Faith Theorem for [Formula: see text]-injective modules are provided. And the Cartan–Eilenberg–Bass Theorem for [Formula: see text]-Noetherian rings is given in terms of [Formula: see text]-injective modules.