Duo Property on the Monoid of Regular Elements

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 [Formula: see text] is right DR if and only if [Formula: see text] is commutative. It is also proved that for a prime number [Formula: see text], the group ring [Formula: see text] of a finite [Formula: see text]-group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] is right DR if and only if it is right duo, and that there exists a group ring [Formula: see text] that is neither DR nor duo when [Formula: see text] is not a [Formula: see text]-group.