Structure of weakly one-sided duo Ore extensions

Marks (J. Algebra280 (2004) 463-471) proved that if the skew polynomial ring R[x;sigma] is left or right duo, then R[x;sigma] is commutative. It is proved that if R[x;sigma] is weakly left (resp., right) duo over a reduced ring R with an endomorphism (resp., a monomorphism) sigma, then R[x;sigma] is commutative. This concludes that a noncommutative skew polynomial ring is not weakly left duo when the base ring is reduced. It is also shown that if R[x;sigma] is weakly left duo then the polynomial ring R[x] is weakly left duo. We next study the structure of the Ore extension R[x;sigma,delta] when it is weakly left or right duo.