A ring (say unital for simplicity) is semiprimitive (or Jacobson semisimple) if its Jacobson radical is trivial, or equivalently it has faithful semisimple module. Semiprimitivity is a Morita invariant, meaning that if $R$ and $S$ are unital rings and the categories of (left) $R$-modules and $S$-modules are equivalent, then $R$ is semiprimitive iff $S$ is semiprimitive. In some sense, this should say that semiprimitivity is a categorical notion, i.e., there should be some way to define semiprimitivity for abelian categories (maybe with some extra structure like enough projectives or something) so that $R$ is semiprimitive iff the category of $R$-modules is semiprimitive. 


The proof of Morita invariance doesn't go through categorical notions. One first has that if $R$ is Morita equivalent to $S$, then $S$ must be isomorphic to the endomorphism ring of a finitely generated projective $R$-module. Therefore, $S$ looks like $eM\_n(R)e$ for some idempotent $e$. If $M$ is a faithful semisimple $R$-module, then one easily check $eM^n$ with the obvious module structure is a faithful semisimple $eM\_n(R)e$-module. 


The reliance on $M\_n(R)$ in this proof (and hence on free $R$-modules) bothers me particularly because free $R$-modules cannot be defined categorically without specifying an underlying set functor.


My question is whether there is some purely categorical definition of semiprimitivity that would make sense for abelian categories (perhaps with some extra properties like enough projectives) and that doesn't required using the ring theoretic definition. In other words, I don't want to say that an abelian category is semiprimitive iff the endomorphism ring of every projective object is semiprimitve in the usual sense.


My motivation is that I have a particular family of rings for which I am trying to decide semiprimitivity. I have shown that the module categories for these rings are equivalent to more geometrically defined abelian categories and would like to use the latter, which I understand better, to check semiprimitivity, 



Is there a purely module theoretic characterization of semiprimitive rings?
I believe this answers your question, although it may not be that enlightening. Basically, you can rephrase the existence of a faithful semisimple $R$-module in category-theoretic language as follows. First, an $R$-module $M$ is faithful iff $M$ is a cogenerator in $\mbox{Mod-}R$ for the subcategory of (finitely-generated) projective modules; i.e., for every nonzero map $f : P \rightarrow P'$ between projectives $P$ and $P'$ there exists a map $g : P' \rightarrow M$ such that $gf \neq 0$. Thus the ring $R$ is semiprimitive iff $\mbox{Mod-}R$ contains a semisimple object $M$ that is a cogenerator for the subcategory of projective objects. 


