We know in a semisimple ring R, for every Rmodule, Noetherian is equivalent to Artinian, my question is:
If for every Rmodule M Noetherian is equivalent to Artinian, can we prove R is a semisimple ring?
We know in a semisimple ring R, for every Rmodule, Noetherian is equivalent to Artinian, my question is: If for every Rmodule M Noetherian is equivalent to Artinian, can we prove R is a semisimple ring? 


Let $R= k[\epsilon]/\epsilon^2$. Then module is Artinian if and only if it is Noetherian if and only if it is finitedimensional. This is clear, since if a module is finitedimensional there can't be an infinite ascending or descending sequence, and if it's infinite dimensional than $M/\epsilon$ is an infinitedimensional vector space, so it has infinite ascending and descending chains. But $R$ is obviously not semisimple. 

