The u,u−1lemma revisited

Abstract

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 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 Prüfer, (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 is Prüfer if and only if (3) holds.