The concept of free modules, Projective modules, injective modules and the like form an important area in module theory. The notion of free fuzzy modules was introduced by Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameri introduced the concept of Projective and injective L-modules. In this paper we give an alternate definition for Projective L-modules. We prove that every free L-module is a Projective L-module. Also we prove that if $\mu\in L(P)$ is a Projective L-module, and if $0\rightarrow\eta\rightarrow\vartheta\rightarrow\mu\rightarrow 0$ is a short exact sequence of L-modules then $\nu\otimes\mu>\vartheta$. Further it is proved that if $\mu\in L(P)$ is a Projective L-module then $\mu$ is a fuzzy direct summand of a free L-module.

Inspired by a recent work of Buchweitz and Flenner, we show that, for a semidualizing bimodule C, C-perfect complexes have the ability to detect when a ring is strongly regular. It is shown that there exists a class of modules which admit minimal resolutions of C-Projective modules.

All finite groups with toroidal or Projective cyclic graphs are classified. Indeed, it is shown that the only finite groups with Projective cyclic graphs are S3 × Z2, D14, QD16 and which all have toroidal cyclic graph too. Also, D16 is characterized as the only finite group whose cyclic graph is toroidal but not Projective

Let A be an abelian category with enough Projective objects and let X be a full subcategory of A. We define Gorenstein Projective objects with respect to X and YX, respectively, where YX={YÎCh (A)|Y is acyclic and ZnYÎX}. We point out that under certain hypotheses, these two Gorensein Projective objects are related in a nice way. In particular, if P (A) ÍX, we show that XÎCh (A) is Gorenstein Projective with respect to YX if and only if Xi is Gorenstein Projective with respect to X for each i, when X is a self-orthogonal class or X is Hom (-, X) -exact. Subsequently, we consider the relationships of Gorenstein Projective dimensions between them. As an application, if A is of finite left Gorenstein Projective global dimension with respect to X and contains an in- jective cogenerator, then we find a new model structure on Ch (A) by Hovey’s results in [14].