Zero-dimensionality and the \(GE_ 2\) of polynomial rings (Q1108386)

A commutative ring R is said to be a \(GE_ 2\)-ring if every invertible \(2\times 2\) matrix over R is a product of elementary and diagonal matrices. By using Pierce-sheaf techniques it is shown that R[X], the polynomial ring over R, is a \(GE_ 2\)-ring provided that R is zero- dimensional. The converse is true and it was proved previously by \textit{H. Chu} [J. Algebra 90, 208-216 (1984; Zbl 0541.20033)] who also proved the above theorem in the case where R is noetherian.