0
$\begingroup$

Let $R$ be a commutative ring such that $\{f_1,...,f_n\}\subseteq R$ generates $R$.

Does this imply that for all integers $N>0$ that $\{f_1^N,...,f_n^N\}$ generates $R$?

I would have guessed not, but can't find a counter example and have seen it used in a proof, so can anyone think of a counter example or proof?

$\endgroup$
6
  • 2
    $\begingroup$ Suppose that $R$ is a commutative ring with unit. For any $n$-tuple of positive integers $k_1, \dots, k_n$, try to show that the ideal generated by the elements $f_1^{k_1}, \dots, f_n^{k_n}$ cannot be contained in any prime ideal. So they generate the ideal $(1) = R$. $\endgroup$ – Andrea Gagna Feb 19 '15 at 10:20
  • $\begingroup$ OK, I see why it was stated as if obvious now, thanks! $\endgroup$ – Robert Chamberlain Feb 19 '15 at 10:21
  • 2
    $\begingroup$ But if the ring does not have a unity I think the statement is false. Indeed one can consider the ring of even integers, which is generated by $2$ but not by $4$. Right? $\endgroup$ – Giovanni De Gaetano Feb 19 '15 at 10:22
  • $\begingroup$ The ring in the proof i was reading does have unity, I should have stated that. $\endgroup$ – Robert Chamberlain Feb 19 '15 at 10:26
  • $\begingroup$ @GiovanniDeGaetano In fact your claim is true (as I've shown here), but the example is wrong. $\endgroup$ – user26857 Feb 19 '15 at 22:31