Tensor, trace and composition

The product in a commutative ring can be thought of in (at least) two ways, which can be seen if one goes to matrices. On the one hand, you have usual matrix multiplication and on the other, you have the tensor product of matrices. These constructions are connected through the trace. These two operations give rise to different constructions, each with its own advantages (but which coincide in the one-dimensional case), and the project will be to compare these, in particular with the aim of clarifying the issue in algebraic K-theory.