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I have a problem. As I've understand it two proper ideals $I$ and $J$ of a ring $R$ is said to be $coprime$ if $I+J=(1)$. For a set of distinct maximal ideals of $R$ say $\{I_i\}$, $0\leq i\leq n$, how can I see that these are pairvise coprime..? $R$ is assumed to be a Noetherian domain. I can't see how to work this out without using the concept of GCD, and that is not a welldefined concept in such a ring.

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Well, if $I$ and $J$ are distinct maximal ideals in a commutative ring with unity $R$, then $I+J$ is also an ideal, and $$ I \subset I+J \subseteq R. $$ The first inclusion is strict, and $I$ is a maximal ideal, therefore $I+J=R=(1)$. Unless I misunderstand the question.

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It is enough to concider two maximal ideals $I$ and $J$. If $I$ and $J$ are distinct and $I+J\neq (1)$, then $I,J\subsetneq I+J\subsetneq R$, contradicting the maximality of $I$ and $J$.

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