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Let $R$ be a $\mathbb{C}$-algebra satisfying the following conditions:

(i) $R \subset \mathbb{C}[x_1,\ldots,x_n]$.

(ii) There exist $a_1,\ldots,a_l \in \mathbb{C}[x_1,\ldots,x_n]$ such that $R=\mathbb{C}[a_1,\ldots,a_l]$, for some $l > n$.

(iii) $R_m$ is a UFD, where $m= \langle a_1,\ldots,a_l \rangle$.

Question. Is $R$ a UFD?

A non-example: $R=\mathbb{C}[x^2,x^3] \subset \mathbb{C}[x]$, $m=\langle x^2,x^3 \rangle$. If I am not wrong, condition (iii) does not hold, since $R$ is of course not a UFD ($x^2x^2x^2=x^3x^3$), so its localization at $m$ is also not a UFD.

Perhaps Kaplansky criterion can help? Also, perhaps the list in wikipedia for equivalent conditions for being a UFD may help.

Any hints and comments are welcome! Thank you.

Edit: Mohan's example $R=\mathbb{C}[x^2-1,x^3-1]$ satisfies conditions (i)+(ii)+(iii) (truly, I am not sure why $R_m$ is a UFD), but $R$ is not a UFD. Therefore, I wish to change condition (iii) to the following stronger condition: $R_m$ is regular, where $m= \langle a_1,\ldots,a_l \rangle$.

If I am not wrong, Mohan's example does not satisfy this new condition, since the minimal number of generators of $m$ is $2$ and the Krull dimension of $R_m$ is $1$.

user237522
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  • Right: neither $R_m$ nor $R$ is a UFD, so you don’t have a case where the three conditions are satisfied without $R$ being a UFD. – Lubin Apr 01 '21 at 03:36
  • Instead of condition (iii), asuume the stronger condition (iv): $R_m$ is regular, $m=\langle a_1,\ldots,a_l \rangle$. Then the minimal number of generators of $m$ equals $\dim(R_m) \leq n < l$. – user237522 Apr 01 '21 at 09:45
  • For any such $R$ you can find $a_i$s satisfying your conditions. So, pick any non UFD $R$ contained in the polynomial ring. – Mohan Apr 01 '21 at 14:10
  • @Mohan, please could you explain "for any such $R$...". Unfortunately, I am not sure I understand what you meant. My purpose is to apply the above result to a ring $R$ which is not known to be a UFD, and to obtain that it is a UFD. – user237522 Apr 01 '21 at 21:10
  • Are you saying that in my 'non-example' $R_m$ is a UFD? – user237522 Apr 01 '21 at 21:17
  • No. $R=\mathbb{C}[x^2-1,x^3-1]$ and localize at the maximal ideal $(x^2-1,x^3-1)$ to get a UFD. – Mohan Apr 02 '21 at 00:08
  • Thank you. Is there an additional condtion to conditions (i)+(ii)+(iii) such that all four conditions imply that $R$ is a UFD? – user237522 Apr 02 '21 at 00:16
  • Any localization of your $R$ at a maximal ideal other than $(x^2,x^3)$ is regular, in particular at $(x^2-1,x^3-1)$. – Mohan Apr 02 '21 at 16:16
  • Thank you. So the new condition (iii) concerning regularity does not help.. – user237522 Apr 03 '21 at 18:57

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