• BALANCED GROUPS AND RINGS

Abstract

A finite group G is balanced if the product of its elements, in some order, or its square is the identity element. We give examples of some balanced groups. Let R be a ring (not necessarily commutative) with a finite cyclic group U of units of order n. It is shown that the product of units is -1 if n is even and it is 1 if n is odd. Let D be an integral domain with a finite number n of units and with ch(D) different from 2. It is shown that the product of units is -1. Let R be a commutative ring with 1 whose characteristic is not 2 and that has a finite group of units U and let the order of U be n. It is shown that n is even. Let R be a commutative ring with 1 which has a finite number of units and let A be the nil radical of R. Then R/A has a finite number of units whose product is 1 or -1 mod A.