On the Whitehead Determinant for Semi-local Rings

I referred her to [2], where Theorem 3.6 asserts that K1(R) is R /Ẽ, where Ẽ is the group generated by (1 + xy)/(1 + yx) with x, y in R and 1 + xy in R and where the last sentence in §3 says that Ẽ is not [R, R] in the case when R = M2(Z/2Z) is the ring of 2 by 2 matrices over a field of two elements (Z is the ring of integers). Moreover, in this case the group K1(R) is trivial while R/[R , R] has order two. Therefore Ẽ 6= [R, R] whenever R has a ring morphism onto M2(Z/2Z), see Theorem 1 below. Recall [1, p. 503] that a ring R is semi-local if and only if the ring R/rad(R) is isomorphic to a finite product of matrix rings over division rings D where rad(R) is the Jacobson radical of R. The Whitehead determinant GLn(R) → K1(R) was introduced for any associative ring R with 1 and any integer n ≥ 1 in [1]. Here is another counter example to (1). Let R = T2(Z/2Z) be the ring of 2 by 2 upper triangular matrices over Z/2Z. In this case, R/rad(R) is isomorphic to (Z/2Z)× (Z/2Z), the multiplicative group R = Ẽ has order two and its commutator subgroup is trivial.