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Let $R$ be a commutative $k$-algebra, where $k$ is a field of characteristic zero.

Could one please give an example of such $R$ which is also:

(i) Not affine (= infinitely generated as a $k$-algebra).

and

(ii) Not an integral domain (= has zero divisors).

My first thought was $k[x_1,x_2,\ldots]$, the polynomial ring over $k$ in infinitely many variables, but unfortunately, it satisfies condition (i) only. It is not difficult to see that it is an integral domain: If $fg=0$ for some $f,g \in k[x_1,x_2,\ldots]$, then there exists $M \in \mathbb{N}$ such that $f,g \in k[x_1,\ldots,x_M]$, so if we think of $fg=0$ in $k[x_1,\ldots,x_M]$, we get that $f=0$ or $g=0$, and we are done.

Thank you very much!

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closed as off-topic by KReiser, user10354138, Saad, user26857, choco_addicted Nov 22 '18 at 6:24

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – KReiser, user10354138, Saad, user26857, choco_addicted
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ This post is missing your thoughts on the problem: what have you tried? This should be relatively straightforwards. $\endgroup$ – KReiser Nov 22 '18 at 0:12
  • $\begingroup$ Truly, I thought to add a remark which says that $k[x_i]_{i \in \mathbb{N}}$ is not an example, since it (ii) is not satisfied. $\endgroup$ – user237522 Nov 22 '18 at 0:15
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    $\begingroup$ Can you think of something that satisfies the first? Can you think of something that satisfies the second? How can you put these two things together to get something that works? $\endgroup$ – user3482749 Nov 22 '18 at 0:16
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    $\begingroup$ @user3482749, thank you. After I have seen the answer, I have also figured out the ideas of your comment. $\endgroup$ – user237522 Nov 22 '18 at 0:21
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    $\begingroup$ In regards to your edited generalization, it is not within the community standards to alter the question so dramatically after you've accepted an answer. You should ask your generalization as a new question. $\endgroup$ – KReiser Nov 22 '18 at 1:07
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For instance, $k[X_n\,:\, n\in\Bbb N]/(X_n^2\,:\, n\in\Bbb N)$: the ring of polynomials in infinitely many variables quotiented by the ideal generated by the square of the variables.

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  • $\begingroup$ Thank you. (I have just edited my question to a slightly less trivial question). $\endgroup$ – user237522 Nov 22 '18 at 1:00

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