Subnormal structure of two-dimensional linear groups over full rings (Q1810150)

We work with commutative rings \(R\) with 1. For any integers \(m\geq 1\) and \(n\geq 2\), B. R. McDonald introduced the concept of \((m,n)\)-full ring. The class of these rings decreases when \(m\) or \(n\) grows. The \((1,2)\)-full rings are the rings satisfying the first stable range condition of H. Bass. Let \(G\) be a subgroup of \(\text{GL}_2(R)\) containing \(\text{SL}_2(R)\). The author describes all subnormal subgroups of \(G\) given that \(R\) is \((1,6)\)-full, \((2,4)\)-full, or \((3,3)\)-full (Theorem 1). When \(R\) is \((1,18)\)-full or \((2,10)\)-full, a more precise description is given in Theorem 2. Note that for subgroups \(G\) of \(\text{GL}_k(R)\) with \(k\geq 3\) containing \(E_k(R)\), a similar description of all subnormal subgroups holds without any restrictions on \(R\).