The Dual of a Strongly Prime Ideal

Let R be a commutative integral domain with quotient field K and let P be a nonzero strongly prime ideal of R. We give several characterizations of such ideals. It is shown that (P : P) is a valuation domain with the unique maximal ideal P. We also study when $P^{?1}$ is a ring. In fact, it is proved that $P^{?1} = (P : P)$ if and only if P is not invertible. Furthermore, if P is invertible, then R = (P : P) and P is a principal ideal of R.