In this paper, we study a special class of Finsler metrics which are defined by a Riemannian metric and a 1-form on a manifold. We find equations that characterize Ricci-flat Douglas metrics among this class.

Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be a multiplicatively closed subset. In this paper, we study [Formula: see text]-Artinian rings and finitely [Formula: see text]-cogenerated rings. A commutative ring [Formula: see text] is said to be an [Formula: see text]-Artinian ring if for each descending chain of ideals [Formula: see text] of [Formula: see text] there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text] Also, [Formula: see text] is called a finitely [Formula: see text]-cogenerated ring if for each family of ideals [Formula: see text] of [Formula: see text] where [Formula: see text] is an index set, [Formula: see text] implies [Formula: see text] for some [Formula: see text] and a finite subset [Formula: see text] Moreover, we characterize some special rings such as Artinian rings and finitely cogenerated rings. Also, we extend many properties of Artinian rings and finitely cogenerated rings to [Formula: see text]-Artinian rings and finitely [Formula: see text]-cogenerated rings.

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In this study, we introduce the concepts of S-prime submodules and S-torsion-free modules, which are generalizations of prime submodules and torsion-free modules. Suppose S ⊆ R is a multiplicatively closed subset of a commutative ring R , and let M be a unital R-module. A submodule P of M with (P :R M) ∩ S = ∅ is called an S-prime submodule if there is an s ∈ S such that am ∈ P implies sa ∈ (P :R M) or sm ∈ P. Also, an R-module M is called S-torsion-free if ann(M) ∩ S = ∅ and there exists s ∈ S such that am = 0 implies sa = 0 or sm = 0 for each a ∈ R and m ∈ M. In addition to giving many properties of S-prime submodules, we characterize certain prime submodules in terms of S-prime submodules. Furthermore, using these concepts, we characterize some classical modules such as simple modules, S-Noetherian modules, and torsion-free modules.

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