Okay, first of all, what is the definition of right finitely generated? I wanted to apply Osofsky's lemma somewhere in my work. The lemma says the following:
If $R$ is a left perfect ring in which $J/J^2$ is right finitely generated, then $R$ is right artinian.
$J = Jac(R)$. I know that $R$ is finitely generated and so is $R/J^2$ (generated by $1+J^2$). Does this mean that $R/J^2$ is both left and right finitely generated? If in my work where I managed to conclude that $J/J^2\cong \bigoplus_{i=1}^n S_i$, for simple right $R/J$-modules $S_i$. Does it imply that $J/J^2$ is finitely generated as a right $R/J$-module or as a right $R$-module?
Huge thanks for any help!