Integrally closed domains

Every UFD is integrally closed, by the rational root argument. A domain \(A\) is integrally closed if and only if \(A_\mfp\) is integrally closed for every prime \(\mfp\); so an integral scheme \(\Spec A\) is normal, in the sense that its local rings are integrally closed, if and only if \(A\) is integrally closed.

The coordinate ring of the cusp, \(A = k[t^2,t^3] \subseteq k[t]\), is not integrally closed: the element \(t = t^3/t^2\) lies in \(K(A)\) and satisfies \(x^2 - t^2 = 0\), so it is integral over \(A\) but not in \(A\), and \(k[t]\) is the integral closure. Geometrically the normalisation \(\Spec k[t] \to \Spec A\) is the map resolving the cusp of \(y^2 = x^3\).

For local rings, regular \(\implies\) UFD \(\implies\) integrally closed, with both implications strict; for a Noetherian local domain of dimension one, the three conditions coincide and characterize DVRs. A regular ring that is not local need not be a UFD: the Dedekind domain \(\ZZ[\sqrt{-5}]\) is regular and not a UFD.