I am trying to prove something about the group of units $T^*$ of the ring of 2X2 upper triangle matrices $T$ over a ring $R$ (with 1 not equal to 0), namely that $T^*$ being commutative implies that $R^*$ is trivial.
I have yet been able to prove that the diagonal elements of matrices in $T^*$ must be from $R^*$ and a bit later even that they have to be in the center of $R^*$. But I got stuck from there.