Nikaido's Theorem

Nikaido's Theorem states that in certain mathematical systems called "commutative rings," if the ring is "local" (has a single maximal ideal) and "Noetherian" (satisfies a finiteness condition), then every element that is "locally invertible" (invertible in every localization) is actually invertible in the entire ring. In simpler terms, under these conditions, if an element behaves like a unit in all localized versions of the ring, it is a unit everywhere. This result helps mathematicians understand how local properties influence the global structure of algebraic systems.