Is it possible to imbed any ring into a semi-simple ring?

Question

As the title states: Is it possible to imbed any ring as a sub-ring of a semi-simple ring? It seems to me that this question breaks down to asking whether or not for any ring $R$, there exists a semi-simple ring, say $S$, in which $R$ and all it's subrings have compliments.

Now, I have no counter-example currently but intuition tells me no and I don't see an obvious way of constructing one.

Answer 1

Since you're talking about complements, I suppose you're talking about "semisimple (Artinian) rings" and not merely rings with zero Jacobson radical.

No: it is not possible. The first thing that occurs to me is that a semisimple Artinian ring is Dedekind finite, and that passes to unital subrings. Dedekind finite means that $xy=1\implies yx=1$.

So, if one chooses a ring which isn't Dedekind finite, it can't be a subring of any Dedekind finite ring (and Artinian rings have that property.)

Answer 2

I'll make a try using counterexample

$R$ left Artinian & $Jac(R)=0$ implies $R$ is semisimple (semisimple as an $R$-module)

So there are rings which are not subrings of semisimple rings.