Duo Property on the Monoid of Regular Elements

Abstract

We study the right duo property on regular elements, and we say that rings with this property are right DR. It is first shown that the right duo property is preserved by right quotient rings when the given rings are right DR. We prove that the polynomial ring over a ring R is right DR if and only if R is commutative. It is also proved that for a prime number p, the group ring KG of a finite p-group G over a field K of characteristic p is right DR if and only if it is right duo, and that there exists a group ring KG that is neither DR nor duo when G is not a p-group.