Decomposition of Steinberg groups (Q1202812)

Let \(A\) be an associative ring with 1, \(n \geq 2\) an integer. In the group \(E_ n(A)\) generated by \(n\text{ by }n\) elementary matrices, we consider the following subgroups: \(U\) (resp. \(L\)) the subgroup of all unit upper (resp. lower) triangular matrices, \(W\) the subgroup of monomial matrices. Theorem 3 asserts that for a commutative ring \(A\), \(E_ n(A) = LWLU\) if and only if \(A\) satisfies the first Bass stable range condition. Theorem 5 asserts that \(E_ n(A) = LWL\) if and only if \(A\) is a division ring. Similar results are obtained for the Steinberg groups \(\text{St}_ n(A)\).