Dedekind domain [SEP] named after
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.
A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a
Prüfer domain
In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.
Examples.
The ring of entire functions on the open complex plane C form a Prüfer domain. The ring of integer valued polynomials with rational number coefficients is a Prüfer domain, although the ring Z["X"]