  A pair of elements $a,b$ in an integral domain $R$ is an idempotent pair if
either $a(1-a) \in bR$, or $b(1-b) \in aR$. $R$ is said to be a PRINC domain if
all the ideals generated by an idempotent pair are principal. We show that in
an order $R$ of a Dedekind domain every regular prime ideal can be generated by
an idempotent pair; moreover, if $R$ is PRINC, then its integral closure, which
is a Dedekind domain, is PRINC, too. Hence, a Dedekind domain is PRINC if and
only if it is a PID. Furthermore, we show that the only imaginary quadratic
orders $\mathbb Z[\sqrt{-d}]$, $d > 0$ square-free, that are PRINC and not
integrally closed, are for $d=3,7$.
