The subgroup generated by the involutions in a compact ring

摘要

IfAis a compact ring with identity andGis the group of units inA, an elementginGis an involution ofAifg2= 1. Let ▵ denote the set of involutions inAand letWbe the subgroup ofGgenerated by ▵. Giveng∈W, the lengthl(g), ofgis the smallest positive integermsuch that there exist. It is shown thatWis compact if and only if there exists a positive integer m such thatl(g),≤mfor allginW. Moreover ifAis a compact semisimple ring or ifAis a compact ring such that 2 is a unit inAand the Jacobson radicalJofAcontains no nonzero algebraic nilpotent element, thenl(g),≤ 4 for allginW. Furthermore in the latter casel(g) = 1 for allginWif and only ifA/Jis isomorphic to a product of finite fields, each having odd characteristic. In general ▵ is either finite or uncountable. IfWis finitely generated, tenWis finite. It is also shown that innis an odd integer greater than 3, thenWis not isomorphic to the dihedral group Dn, However, for each prime p greater than 3, there exists a finite ringAsuch thatWis isomorphic toD2pFinally those compact rings for whichW=Gare considered. In particular it is shown that, up to isomorphism, the ring of integers module 3 is the unique local compact ring for whichW=Gand for which 2 is a unit.