Classes of generalized $*$-modules

Let R be an associative ring with identity element. Following Menini and Orsatti, we say that a module P∈Mod-R is a∗-module provided the functors HomR(P,−) and −⊗R′P induce an equivalence between Gen(PR) and Cog(P∗R′), where R′=End(PR) and P∗=HomR(P,Q) for an injective cogenerator Q of Mod-R. Let λ be a nonzero cardinal number. An R-module P is said to be a∗λ-module if P is finitely generated and P satisfies the condition C(κ) for all κ<λ, where C(κ) denotes the following assertion: "For every submodule M of P(κ), the condition M∈Gen(PR) is equivalent to the injectivity of the canonical morphism ExtR(P,M)→ExtR(P,P(κ)).'' A module is a ∗-module if and only if it is a∗λ-module for all λ. The aim of this paper is the investigation of the class of all ∗-modules by means of the classes Sλ of all∗λ-modules. In the last section of the paper the authors answer a question of Menini, by showing that the chain Sλ, λ>ℵ0, is strictly decreasing in general.