On mutual commutants of generalized carpet subgroups (Q1586968)

Let \(R\) be a commutative ring containing a unit. A set of ideals \({\mathcal A}=\{{\mathcal A}_{ij}\mid i,j\in\mathbb{Z}\}\) of \(R\) is called a carpet of ideals if \({\mathcal A}_{ik}{\mathcal A}_{kj}\subseteq{\mathcal A}_{ij}\). The sets \(G({\mathcal A})=\{g\in\text{GL}_n(R)\mid g_{ij}\cong\delta_{ij}\bmod{\mathcal A}_{ij}\}\) and \(S({\mathcal A})=G({\mathcal A})\cap\text{SL}_n(R)\) are called the generalized carpet subgroup and the special carpet subgroup. The notion and properties of carpet subgroups in linear groups and Chevalley groups can be found in [\textit{Yu.~I.~Merzlyakov}, Algebra Logika 3, No. 4, 49-59 (1964; Zbl 0244.20048)]. The only case in which the structure of carpet subgroups still remains unknown is as follows: \(n=2\) and the element 2 of \(R\) is not invertible. In the article under review, some connections between commutators of carpet subgroups (in the exceptional case) and the field \(R/2R\) are found.