Let it 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 ∈ K is the root of a polynomial f ∑ R [X] with some coefficient a unit, then u or u-1 ∈ R. A globalization states that for R integrally closed, if [image omitted] is the root of f ∈ R [X] with Af invertible, then (a, 6) 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 ∑ K is the root of a quadratic polynomial f ∑ R [X] with some coefficient a unit, and (3) every u ∈ K is the root of a polynomial f ∈ R [X] with Af invertible. Moreover, for any integral domain R, the integral closure [image omitted] is Prufer if and only if (3) holds.
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