\(FP\)-Gorenstein cotorsion modules.
An \(R\)-module \(N\) is defined to be FP-Gorenstein cotorsion if \(\text{Ext}^1_R(F,N)=0\) for all finitely presented Gorenstein flat \(R\)-modules \(F\). This concept clearly generalizes the concepts of injectivity, FP-injectivity and Gorenstein cotorsion modules. Properties of FP-Gorenstein cotorsion modules are first investigated. The class of FP-Gorenstein cotorsion modules is shown to be closed under direct products, finite direct sums, direct summands, extensions by FP-Gorenstein cotorsion, pure submodules and, if the ring is coherent, under direct limits. Over coherent rings, every \(R\)-module is shown to have a surjective FP-Gorenstein cotorsion cover and a special FP-Gorenstein cotorsion preenvelope. The FP-Gorenstein cotorsion dimension (\(\text{FP-Gcd}(M)\)) of an \(R\)-module and of a ring \(R\) are next defined. If \(R\) is a coherent ring, \(\text{FP-Gcd}(M)\) and the global FP-Gorenstein cotorsion dimension are shown to equal 0 or \(\infty\) and a perfect, hereditary cotorsion pair is formed. Several properties follow for specific rings. For example: A Noetherian ring \(R\) is shown to be QF if and only if every FP-Gorenstein cotorsion \(R\)-module is injective and a coherent ring \(R\) is proved to be an FC-ring if and only if every FP-Gorenstein cotorsion \(R\)-module is FP-injective. If every FP-Gorenstein cotorsion \(R\)-module is Gorenstein cotorsion, \(R\) coherent, then \(R\) is left perfect. For \(R\) an \(n\)-FC ring with \(n\geq 0\), the following are equivalent: (i) \(\text{wD}(R)<\infty\); (ii) Every finitely presented Gorenstein flat \(R\)-module is projective; (iii) Every \(R\)-module is FP-Gorenstein cotorsion; (iv) Every quotient of an FP-Gorenstein cotorsion module is FP-Gorenstein cotorsion; (v) Every submodule of an FP-Gorenstein cotorsion \(R\)-module is FP-Gorenstein cotorsion.
scientific article; zbMATH DE number 6305713