My apologies if this question is too vague:
If one has a commutative ring $R$, a projective $R$-module $M$ and an endomorphism $\Phi$ of $M$, one can take the usual determinant of $\Phi$ and everything works fine. This is not the case when $R$ is non-commutative, but there are some workarounds when $R$ is semisimple:
Every simple component (say $R_i$) is a CSA over its center Z$(R_i)$, which is a field. Then one can take a splitting field $E_i$ for $R_i$ (e.g. a finite extension of Z$(R_i)$) so that $R_i \otimes_{Z(R_i)}E_i$ is isomorphic to some ring of matrices over the field $E_i$ and here we can take determinants again. So you can find the so called reduced norm of $\Phi$ by looking at each simple component of $R$ and then taking a product.
If you do this procedure on an already commutative ring, how are the reduced norm and the standard determinant related?
And now the vague part: how much do we know in general about 'translating' identities involving determinants to non-commutative settings?