Dedekind domain

English Etymology Named after German mathematician Richard Dedekind (1831–1916). Noun en algebra ring theory An integral domain in which every proper ideal factors into a product of prime ideals which is unique (up to permutations). en It can be proved that a Dedekind domain (as defined above) is equivalent to an integral domain in which every proper fractional ideal is invertible. en 1971 Max D. Larsen; Paul J. McCarthy Multiplicative Theory of Ideals Elsevier (Academic Press) https://books.google.com.au/books?id=jPCin77MOjQC&pg=PA201&dq=%22Dedekind+domain%22%7C%22Dedekind+domains%22&hl=en&sa=X&ved=0ahUKEwi05rKO29jgAhUCMXwKHWzxAxAQ6AEIRDAF#v=onepage&q=%22Dedekind%20domain%22%7C%22Dedekind%20domains%22&f=false 201 In this chapter we shall study several of the important classes of rings which contain the class of Dedekind domains. en 2007 Leonid Kurdachenko; Javier Otal; Igor Ya. Subbotin Artinian Modules over Group Rings Springer (Birkhäuser) https://books.google.com.au/books?id=u8wbIsbb6J0C&pg=PA55&dq=%22Dedekind+domain%22%7C%22Dedekind+domains%22&hl=en&sa=X&ved=0ahUKEwi05rKO29jgAhUCMXwKHWzxAxAQ6AEISTAG#v=onepage&q=%22Dedekind%20domain%22%7C%22Dedekind%20domains%22&f=false 55 As we can see every principal ideal domain is a Dedekind domain. en 2007 Anthony W. Knapp Advanced Algebra Springer (Birkhäuser) https://books.google.com.au/books?id=25JfJAgqC8sC&pg=PA266&dq=%22Dedekind+domain%22%7C%22Dedekind+domains%22&hl=en&sa=X&ved=0ahUKEwi05rKO29jgAhUCMXwKHWzxAxAQ6AEIZzAL#v=onepage&q=%22Dedekind%20domain%22%7C%22Dedekind%20domains%22&f=false 266 Let us recall some material about Dedekind domains from Chapters VIII and IX of Basic Algebra. A Dedekind domain is a Noetherian integral domain that is integrally closed and has the property that every nonzero prime ideal is maximal. Any Dedekind domain has unique factorization for its ideals. Usage notes For a list of equivalent definitions, see Dedekind domain#Alternative definitions Synonyms integral domain whose prime ideals factorise uniquely en Dedekind ring Hypernyms integral domain whose prime ideals factorise uniquely en Noetherian domain Derived terms en almost Dedekind domain Translations integral domain whose prime ideals factorise uniquely French: fr anneau de Dedekind m German: de Dedekindring m Italian: it dominio di Dedekind m, it anello di Dedekind m Polish: pl pierścień Dedekinda m Portuguese: pt domínio de Dedekind m, pt anel de Dedekind m Further reading Ideal class group Unique factorization domain Dedekind ring on Encyclopedia of Mathematics Dedekind Ring on MathWorld Wolfram MathWorld en eponyms