Given a unital ring $R$, it's easy to form the ring $M_n(R)$ of square $n\times n$ matrices over $R$. Any such matrix has a trace, which is simply the sum along the diagonal.
However, I've come to realize that it's not so simple for $R$ non-commutative. In that case, we take the sum of the diagonal elements, which gives us an $r\in R$, and then we project it into $R/[R, R]$, the ring $R$ divided out by all commutators. Why is that?
Is the trace not as well-behaved as we would like it in the non-commutative case? Or is it rather that any time you would want to take the trace (such as, for instance, when studying $K_0(R)$), it just so happpens to be more natural to see it as an element in $R/[R, R]$ and therefore it's just been made a convension that that's where traces should live?