(algebra, ring theory) An integral domain in which every proper ideal factors into a product of prime ideals which is unique (up to permutations).
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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.
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In this chapter we shall study several of the important classes of rings which contain the class of Dedekind domains.
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1971, Max D. Larsen, Paul J. McCarthy, Multiplicative Theory of Idealshttps://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, Elsevier (Academic Press), page 201:
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As we can see every principal ideal domain is a Dedekind domain.
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2007, Leonid Kurdachenko, Javier Otal, Igor Ya. Subbotin, Artinian Modules over Group Rings, Springer (Birkhäuser), page 55:
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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.
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2007, Anthony W. Knapp, Advanced Algebrahttps://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, Springer (Birkhäuser), page 266: