The u, u(-1) Lemma revisited

Let R be an integral domain with quotient field K. The u, u(-1) Lemma states that if R is integrally closed and quasilocal and if u is an element of K is the root of a polynomial f is an element of R[X] with some coefficient a unit, then u, or u(-1) is an element of R. A globalization states that for R integrally closed, if u = a/b is the root of f is an element of R [X] with A(f) invertible, then (a, b) is invertible. We prove the converse of both results and show that for R integrally closed, the following are equivalent: (1) R is Prufer, (2) every u is an element of K is the root of a quadratic polynomial f is an element of R[X] with some coefficient a unit, and (3) every u is an element of K is the root of a polynomial f is an element of R[X] with A(f) invertible. Moreover, for any integral domain R, the integral closure (R) over bar is Prufer if and only if (3) holds.