Arlinghaus, F. A.; Vaserstein, L. N. Work of Pere Menal on normal subgroups. (English) Zbl 0790.20064 Publ. Mat., Barc. 36, No. 2A, 389-400 (1992). 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. Reviewer: Alexander W. Mason (Glasgow) Cited in 1 Document MSC: 20G35 Linear algebraic groups over adèles and other rings and schemes 20H25 Other matrix groups over rings 20E15 Chains and lattices of subgroups, subnormal subgroups 20E07 Subgroup theorems; subgroup growth 20-03 History of group theory 01A60 History of mathematics in the 20th century Keywords:two-dimensional general linear groups; subgroup structure of \(GL_ 2A\); elementary matrices; \(E_ 2A\)-normalized subgroups; von Neumann regular ring; Banach algebra; stable range 1 ring; local ring; quasi-ideals PDFBibTeX XMLCite \textit{F. A. Arlinghaus} and \textit{L. N. Vaserstein}, Publ. Mat., Barc. 36, No. 2A, 389--400 (1992; Zbl 0790.20064) Full Text: DOI EuDML