Work of Pere Menal on normal subgroups
The authors survey the work of the late Pere Menal on the subgroup structure of \(\mathrm{GL}_ 2A\) for rings \(A\) of various types. For a large class of rings \(A\), which includes all commutative, von Neumann regular and stable range 1 rings, together with all Banach algebras, it is known that, when \(n \geq 3\), the subgroups of \(\mathrm{GL}_ nA\) normalized by \(E_ nA\) (the subgroup generated by the elementary matrices) can be completely classified in terms of the (two-sided) \(A\)-ideals. For anything like such a classification to extend to the case \(n = 2\) further restrictions usually have to be imposed on \(A\). (Any classification of this type appears to fail, for example, when \(A = \mathbb Z\).) Menal (in collaboration with the second author) has classified the \(E_ 2A\)-normalized subgroups of \(\mathrm{GL}_ 2A\) when (i) \(A\) is a von Neumann regular ring, (ii) \(A\) is a Banach algebra or a stable range 1 ring containing \(1/2\), and (iii) \(A\) is a local ring, where \(A/\text{rad }A\) has at least 4 elements. In case (i) the classification involves the \(A\)-ideals. For (ii), (iii) the classification involves the quasi-ideals of \(A\). The principal way in which these results improve on the (many) previously known special cases is that in nearly all of the latter the ring is assumed to be commutative.
scientific article; zbMATH DE number 203304