Subgroups of GL₁(R) for Local Rings R

Abstract

Let R be a local ring, with maximal ideal m , and residue class division ring R/ m  = D. Denote by R* = G L ₁(R), the group of units of R. Here we investigate some algebraic structure of subnormal and maximal subgroups of R*. For instance, when D is of finite dimension over its center, it is shown that finitely generated subnormal subgroups of R* are central. It is also proved that maximal subgroups of R* are not finitely generated. Furthermore, assume that P is a nonabelian maximal subgroup of R* such that P contains a noncentral soluble normal subgroup of finite index, it is shown that D is a crossed product division algebra.

Key Words:

• Local ring
• Subnormal subgroup
• Maximal subgroup

AMS (2000) Subject Classification:

• 15A33
• 16K

Acknowledgments

The second author is indebted to the Research Council of Sharif University of Technology and the Abdus Salam International Centre for Theoretical Physics for support.

Notes

^#Communicated by C. Pedrini.