Let $R$ be a ring. A left $R$-module $M$ is called an FC-module if $M^{+}={Hom}_{\mathbb{Z}}(M,\mathbb{Q}/\mathbb{Z})$ is a flat right $R$-module. In this paper, some homological properties of FC-modules are given. Let $n$ be a nonnegative integer and ${\mathrm{ℱ𝒞}}_n$ the class of all left $R$-modules $M$ such that the flat dimension of $M^{+}$ is less than or equal to $n$. It is shown that $({}^{\perp }\left({\mathrm{ℱ𝒞}}_n^{\perp }\right),{\mathrm{ℱ𝒞}}_n^{\perp })$ is a complete cotorsion pair and if $R$ is a ring such that $fd\left({\left({}_{R}R\right)}^{+}\right)\le n$ and ${\mathrm{ℱ𝒞}}_n$ is closed under direct sums, then $({\mathrm{ℱ𝒞}}_n,{\mathrm{ℱ𝒞}}_n^{\perp })$ is a perfect cotorsion pair. In particular, some known results are obtained as corollaries....

We present an alternative way of measuring the Gorenstein projective (resp., injective) dimension of modules via a new type of complete projective (resp., injective) resolutions. As an application, we easily recover well known theorems such as the Auslander-Bridger formula. Our approach allows us to relate the Gorenstein global dimension of a ring R to the cohomological invariants silp(R) and spli(R) introduced by Gedrich and Gruenberg by proving that leftG-gldim(R) = maxleftsilp(R), leftspli(R),...

The strong global dimension of a finite dimensional algebra A is the maximum of the width of indecomposable bounded differential complexes of finite dimensional projective A-modules. We prove that the strong global dimension of a finite dimensional radical square zero algebra A over an algebraically closed field is finite if and only if A is piecewise hereditary. Moreover, we discuss results concerning the finiteness of the strong global dimension of algebras and the related problem on the density...