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Abramo
Abramo
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Let $R$ be a number ring and a Dedekind domain. We have the following result:

For every ideal $I\subset R$ $$ I = \bigcap_P I_P $$ where $I_P$ denotes the localization of $I$ at $P$ and the intersection is taken over all the prime ideals $P$ of $R$.

My question is: Can we deduce from this that if $I_P$ is principal in $R_P$ for every prime ideal $P$ of $R$, then $I$ itself is principal? Or stated differently, is being principal a local property?

Let $R$ be a number ring. We have the following result:

For every ideal $I\subset R$ $$ I = \bigcap_P I_P $$ where $I_P$ denotes the localization of $I$ at $P$ and the intersection is taken over all the prime ideals $P$ of $R$.

My question is: Can we deduce from this that if $I_P$ is principal in $R_P$ for every prime ideal $P$ of $R$, then $I$ itself is principal? Or stated differently, is being principal a local property?

Let $R$ be a number ring and a Dedekind domain. We have the following result:

For every ideal $I\subset R$ $$ I = \bigcap_P I_P $$ where $I_P$ denotes the localization of $I$ at $P$ and the intersection is taken over all the prime ideals $P$ of $R$.

My question is: Can we deduce from this that if $I_P$ is principal in $R_P$ for every prime ideal $P$ of $R$, then $I$ itself is principal? Or stated differently, is being principal a local property?

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Abramo
Abramo
  • 251
  • 3
  • 9

Is being principal a local property?

Let $R$ be a number ring. We have the following result:

For every ideal $I\subset R$ $$ I = \bigcap_P I_P $$ where $I_P$ denotes the localization of $I$ at $P$ and the intersection is taken over all the prime ideals $P$ of $R$.

My question is: Can we deduce from this that if $I_P$ is principal in $R_P$ for every prime ideal $P$ of $R$, then $I$ itself is principal? Or stated differently, is being principal a local property?