On the Whitehead determinant for semi-local rings (Q1763747)

Let \(R\) be a semi-local ring, commutative or not. So, \(R\) modulo its Jacobson radical is a finite product of matrix rings over division rings. The author addresses the question whether the abelianization of \(R^*=\text{GL}_1(R)\) maps isomorphically to \(K_1(R)\). In other words, one wants to know if it maps isomorphically to the the abelianization of \(\text{GL}_n(R)\) for \(n\geq3\). He shows that this happens when none of the matrix rings is isomorphic with \(M_2(\mathbb Z/2\mathbb Z)\) and no more than one of the matrix rings has order 2. He also gives examples showing the need for some restrictions.