Let $(R, \mathfrak{m})$ be a Noetherian local ring and $M$, $N$ be two finitely generated $R$-modules. In this paper it is shown that $R$ is a Cohen-Macaulay ring if and only if $R$ admits a non-zero Artinian $R$-module $A$ of finite projective dimension; in addition, for all such Artinian $R$-modules $A$, it is shown that $\mathrm{pd}_R\, A=\dim R$. Furthermore, as an application of these results it is shown that$$\pdd H^i_{{\frak p}R_{\frak p}}(M_{\frak p}, N_{\frak p})\leq \pd H^{i+\dim R/{\frak p}}_{\frak m}(M,N)$$for each ${\frak p}\in \mathrm{Spec} R$ and each integer $i\geq 0$. This result answers affirmatively a question raised by the present authors in [13].

Let R be a commutative Noetherian ring, M be a finitely generated R-module and a be an ideal of R. For an arbitrary integer k ,􀀀, 1, we introduce the concept of k-projective dimension of M denoted by k-pdRM. We show that the finite k-projective dimension of M is at least k-depth(a, R)􀀀, k-depth(a, M). As a generalization of the Intersection Theorem, we show that for any finitely generated R-module N, in certain conditions, k-pdRM is nearer upper bound for dimN than pdRM. Finally, if M is k-perfect, dimN ,k-gradeM that generalizes the Strong Intersection Theorem.

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.

In this paper, a new homological dimension of modules, co-presented dimension, is defined. We study some basic properties of this homological dimension. Some ring extensions are considered, too. For instance, we prove that if S  R is a finite normalizing extension and SR is a projective module, then for each right S-module MS, the cop-resented dimension of MS does not exceed the copresented dimension of HomR(S, M).

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

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.