A ring R (with unit element) is
called a duo ring if every one-sided ideal is two-sided. This is equivalent with the
existence of elements r′ and r′′ in R with rs = sr′,sr = r′′s for elements r,s in R. We
will discuss in this note the following three problems: (A) Is the localization at a
prime ideal P of a duo ring again a duo ring? (B) Is in a duo ring the P-component
of zero equal to the right (left) P-component of zero? (C) Is in a noetherian duo
domain the semi group of ideals (under multiplication) commutative? The answer
to all three questions is “no” in general, but “yes” for (A) and (B) in the
noetherian case, and “yes” for (C) if R is integrally closed in its division ring of
quotients.
