Can somebody give me example of ring $R$ such that $R/I$ is Noetherian but $R$ is not Noetherian ring? $I$ is finitely generated ideal of $R$.

Also please search example such that $I$ is not nilpotent because if $I$ is nilpotent then R become Notherian ring.


closed as off-topic by Shaun, anomaly, Claude Leibovici, Mostafa Ayaz, Gibbs Sep 25 '18 at 13:21

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    $\begingroup$ $R$ is any non-Noetherian ring, and $I = R$ works... $\endgroup$ – darij grinberg Sep 24 '18 at 15:42
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    $\begingroup$ Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. $\endgroup$ – Shaun Sep 24 '18 at 15:55
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    $\begingroup$ $R$ is finitely generated as an iedal. $\endgroup$ – darij grinberg Sep 24 '18 at 15:55
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    $\begingroup$ A non-Noetherian ring $R$ with finitely generated maximal ideals (and $I$ any of those) works. For instance, this and these. $\endgroup$ – Saucy O'Path Sep 24 '18 at 15:56
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    $\begingroup$ @SaucyO'Path: The zero ring has a unity. It's just zero :) $\endgroup$ – darij grinberg Sep 24 '18 at 16:21

$R=\prod_{i\in \mathbb N}F_2$, with $I=(0,1,1,\ldots)R$.

$R/I$ is the field $F_2$ and $I$ is principal.

  • $\begingroup$ How $R/I$ is $\mathbb{F_2}$ ? $\endgroup$ – maths student Sep 24 '18 at 17:08
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    $\begingroup$ @Ninjahatori Just look at the quotient. $I=\{0\}\times F_2\times F_2\times\ldots $. $\endgroup$ – rschwieb Sep 24 '18 at 17:16
  • $\begingroup$ Right Thank you sir $\endgroup$ – maths student Sep 24 '18 at 17:17
  • $\begingroup$ Sir, can you explain me $2\mathbb{Z_(2)} \times \mathbb{Q} $ why this is not -notherian and why ideal of this is principal? I guess non noetherian beacause since it is not finitely generated but how to write it properly I don't know? $\endgroup$ – maths student Sep 24 '18 at 17:21

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