# matrices – Can all finite dimensional non commutative algebras be embedded into matrix rings?

Suppose I have a finite (non-)commutative ring $$R/k$$ (over a field $$k$$ of char $$0$$) with a linear “trace” function $$t: R to k$$. Can I find square matrices $$A_1,dots,A_n$$ (of some dimension $$r$$) so that I have an embedding $$f: R to M_r(k)$$ compatible with the trace functions on both sides?

One restriction I can see for the trace function on $$R$$ is that it should be invariant under cyclic permutations : $$t(a_1a_2dots a_n) = t(a_2dots a_na_1)$$. Is this the only restriction?