%0 Journal Article %T ON THE PROJECTIVE DIMENSION OF ARTINIAN MODULES %J Journal of Algebraic Systems %I Shahrood University of Technology %Z 2345-5128 %A Irani, Y. %A Bahmanpour, K. %A Ghasemi, Gh. %D 2021 %\ 09/01/2021 %V 9 %N 1 %P 13-20 %! ON THE PROJECTIVE DIMENSION OF ARTINIAN MODULES %K projective dimension %K flat dimension %K injective dimension %K generalized localcohomology module %K local cohomology module %R 10.22044/jas.2020.9439.1460 %X 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]. %U https://jas.shahroodut.ac.ir/article_2049_98e21e379801f667abcbe8f1b5044ff5.pdf