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

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.)