Let \(M\) be a commutative monoid, \(R\) a commutative ring and \(\dim R\) its Krull dimension. For a natural number \(c\) denote by \(c_{*}\) the group endomorphism on \(\text{GL}_{r}(R[M])\) or \(K_{2,r}(R[M])\) induced by the monoid endomorphism \(M\rightarrow M\), \(m\rightarrow m^{c}.\) Here \(K_{2,r}(-)\) is the Milnor’s \(r\)th unstable \(K_2\).
The main result of the paper is the following theorem 1.1: Let \(M\) be a commutative cancellative torsion free monoid without nontrivial units, \(c\geq 2\) a natural number; \(R\) a commutative regular ring and \(k\) a field. Then:
(a) For any element \(z\in K_{2,r}(R[M])\), \(r\geq \max(5, \dim R+3)\) there exists \(j_{z}\geq 0\) such that
\[ (c^{j})_{*}(z)\in K_{2,r}(R)=K_2(R), \qquad j\geq j_{z}. \]
(b) For any matrix \(A\in \text{GL}_{r}(R[M])\), \(r\geq \max(3, \dim R+2),\) there exists an integer number \(j_{A}\geq 0\) such that
\[ (c^{j})_{*}(A)\in E_{r}(R[M])\text{GL}_{r}(R), \qquad j\geq j_{A}. \]
(c) There is an algorithm which finds for any matrix \(A\in \text{SL}_{r}(k[M])\), \(r\geq 3,\) an integer number \(j_{A}\geq 0\) and a factorization of the form:
\[ (c^{j})_{*}(A)={\prod}_{k} \, e_{p_{k}q_{k}}({\lambda}_{k}), \qquad e_{p_{k}q_{k}}({\lambda}_{k}) \in E_{r}(k[M]). \]
The proof uses an interesting inductive process which the author calls “pyramidal descent”.