I was asked to show. Let $R$ an Artinian ring and $M$ a left $R$-module finitely generated. Show that $M$ admits a projective cover.
My Attempt: I take any $L$ $R$-module projective finitely generated and an epimorphism $f:L \longrightarrow M$. Then I considered $\mathcal{S}:=\left\{N\subseteq \ker f \mid N \text{ is sub-module of } L \text{ and } f_{N}:L/N\longrightarrow M \text{ is essential epimorphism} \right\} $, then $\mathcal{S}\neq\emptyset$ ($\ker f\in\cal{S}$) and like $L$ is an Artinian $R$-module, exist $X$ minimal with this property, now, I'm trying to prove that $(L/X,f_X)$ is the proyective cover of $M$, but I'm stuck showing that $L/X$ is projective. Any hint?, Thank for your help guys.