Valuations and Prufer Rings

Abstract

The word ring is used to mean commutative ring. Just as valuations on fields are used to study domains, so valuations on rings can be used to study rings; these rings need not have units [12]. We introduce slightly weaker conditions than having identity in order to get a more general theory. A Prufer ring A is one in which every finitely generated regular ideal is invertible. If we replace invertibility in the total quotient ring K, by invertibility in a ring R where A ⊆ R ⊆ K we get an R-Prufer ring. These rings do occur, for example the Witt ring of a non-Pythagorean field or a ring of bounded continuous functions.