The subgroup generated by the involutions in a compact ring

Abstract

If A is a compact ring with identity and G is the group of units in A, an element g in G is an involution of A if g 2 = 1. Let ▵ denote the set of involutions in A and let W be the subgroup of G generated by ▵. Given g ∊ W, the lengthl (g), of g is the smallest positive integer m such that there exist . It is shown that W is compact if and only if there exists a positive integer m such that l(g),≤ m for all g in W. Moreover if A is a compact semisimple ring or if A is a compact ring such that 2 is a unit in A and the Jacobson radical J of A contains no nonzero algebraic nilpotent element, then l(g),≤ 4 for all g in W. Furthermore in the latter casel(g) = 1 for all g in W if and only if A/J is isomorphic to a product of finite fields, each having odd characteristic. In general ▵ is either finite or uncountable. If W is finitely generated, ten W is finite. It is also shown that in n is an odd integer greater than 3, then W is not isomorphic to the dihedral group Dn, However, for each prime p greater than 3, there exists a finite ring A such that W is isomorphic to D 2p Finally those compact rings for which W = G are considered. In particular it is shown that, up to isomorphism, the ring of integers module 3 is the unique local compact ring for which W = G and for which 2 is a unit.