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Is there an integral domain $R$ with a prime ideal $\mathfrak{p}$ of height $1$ which is not a principal ideal?

user8463524
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  • What have you tried? Are you familiar with Dedekind domains which are not principal ideal domains? – Mohan Feb 05 '16 at 21:21

1 Answers1

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It is proved here that the ideal $\mathfrak p=(2,1+\sqrt{-5})$ in $\mathbb Z[\sqrt{-5}]$ is not principal. This is a maximal ideal of height one.

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user26857
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  • I am confused: $(0)$ a prime ideal in this ring since it is a domain. There is a chain $(0)\subsetneq (2)\subsetneq \mathfrak{p}$ of ideals. Your answer implies that $(2)$ is not a prime ideal which means that $2$ is not a prime element in this ring. Is this true? – user8463524 Feb 05 '16 at 21:58
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    @jeffrey Please read this. – user26857 Feb 05 '16 at 22:12