# Example of a finitely generated ideal $I$ with $R/I$ is Noetherian and $R$ not Noetherian. [closed]

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, GibbsSep 25 '18 at 13:21

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• $R$ is any non-Noetherian ring, and $I = R$ works... – darij grinberg Sep 24 '18 at 15:42
• 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. – Shaun Sep 24 '18 at 15:55
• $R$ is finitely generated as an iedal. – darij grinberg Sep 24 '18 at 15:55
• A non-Noetherian ring $R$ with finitely generated maximal ideals (and $I$ any of those) works. For instance, this and these. – Saucy O'Path Sep 24 '18 at 15:56
• @SaucyO'Path: The zero ring has a unity. It's just zero :) – 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.
• How $R/I$ is $\mathbb{F_2}$ ? – maths student Sep 24 '18 at 17:08
• @Ninjahatori Just look at the quotient. $I=\{0\}\times F_2\times F_2\times\ldots$. – rschwieb Sep 24 '18 at 17:16
• 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? – maths student Sep 24 '18 at 17:21