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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: