Is $\mathbb{Z}\Bigl[\frac{1}{2},\frac{1}{3}\Bigr]$ a Dedekind Domain? Can anyone help me with a detailed reasoning?
Is $\mathbb{Z}\Bigl[\frac{1}{2},\frac{1}{3}\Bigr]$ a Dedekind domain?
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abstract-algebra
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0There are *many* definitions and/or characterizations of Dedekind domains. Which do you have available? – 2012-07-24
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0@Bill: Noetherian Integrally closed domain of Krull Dimension $1$. I am also familiar with the definition that every non zero ideal factors as prime ideals and every fractional ideal is invertible, though I don't consider myself handling these definitions very well:( – 2012-07-24
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3Hint: [it is a PID](http://math.stackexchange.com/q/137876/242) $\Rightarrow$ Dedekind. Generally localizations preserves PIDs. $\ \ $ – 2012-07-24
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0@Bill: You mean to say this: We know, that $\mathbb{Z}$ is a P.I.D, so localizing gives $\mathbb{Z}\Bigl[\frac{1}{2},\frac{1}{3}\Bigr]$ as a P.I.D, and since it's a P.I.D its is a Dedikind domain. – 2012-07-24
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0That's one route. – 2012-07-24