Is it possible that an ideal I in an integral domain D is contained in only finitely many maximal ideals but each element of I is contained in infinitely many maximal ideals? I am quit sure that it is possible but I need an example. If some one have in mind, please share.
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2Take $D=\mathbb{C}[X,Y]$, $I=(X,Y)$. – Aug 23 '14 at 06:58
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Thanks for your reply. But note that each maximal ideal of $\mathbb{C}[X,Y]$ is of the form $(X-a,Y-b)$ for some a,b $\in$ $\mathbb{C}$. So it follows $(X,Y)$ is the only maximal ideal containing $X$. Similarly for $Y$. I need an example in which each element of ideal $I$ contained in infinitely many maximal ideals. – Aug 23 '14 at 07:09
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What about $(X,Y-1)$? – Aug 23 '14 at 07:21
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ooooh. Thanks a lot. I was blind at that point. Thank you so much – Aug 23 '14 at 07:22