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Let $R$ be a noetherian ring, and let $I$ be a proper ideal in $R$. If $I$ is generated by $n$ elements, we have by Krull's Principal Ideal Theorem that the height of $I$ is at most $n$. Is it true that $\operatorname{dim}R/I \ge \operatorname{dim}R - n$?

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  • $\begingroup$ By definition . $\endgroup$ – wxu May 22 '12 at 1:03
  • $\begingroup$ So if $\mathfrak p$ is a minimal prime over $I$... then this prime corresponds to a prime ideal in $R / I$. I'm not quite sure why I can extend this chain to length $\operatorname{dim}R - n $... $\endgroup$ – Paul Slevin May 22 '12 at 1:14
  • $\begingroup$ Sorry, I make a mistake. $\endgroup$ – wxu May 22 '12 at 1:22
  • $\begingroup$ Edited to make question clearer to future viewers $\endgroup$ – Paul Slevin May 22 '12 at 9:17
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The answer is no. Let $R=\mathbb{Z}_{(2)}[X]$, and $I=(2X-1)$, where $\mathbb{Z}_{(2)}$ is the ring $\mathbb{Z}$ localizing at the prime ideal $(2)$. Then $R/I$ is a field.

Another cheap counter-example is $R=k\times \mathbb{Z}[X]$ where $k$ is any field. Let $I$ generated by $(0,1)$, i.e., $I=\{0\}\times \mathbb{Z}[X]$. Then $R/I$ is a field. However $\dim R=2$.

It is true for Noetherian local ring $(R,\mathfrak{m})$. Suppose $\dim R/I=r$. Then we can find $x_1,\ldots,x_r$ whose image generating an ideal of definition of $R/I$, that is, $\mathfrak{m}/I$ is minimal over $(x_1,\ldots,x_r)/I$. Say $y_1,\ldots,y_n$ generate $I$, then $\mathfrak{m}$ is minimal over $(x_1,\ldots,x_r,y_1,\ldots,y_n)$, so $\dim R\leq r+n=\dim R/I+n$.

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  • $\begingroup$ So the statement ". For R Noetherian, the dimension of R / I is, by Krull's principal ideal theorem, at least dim R − n" is incorrect on the wikipedia page for "commutative ring" $\endgroup$ – Paul Slevin May 22 '12 at 1:49
  • $\begingroup$ Yes. The counter-example is given as above. $\endgroup$ – wxu May 22 '12 at 1:52
  • $\begingroup$ The reason I was asking is because I wanted to show that $\mathbb Z [x,y] / \langle f \rangle $ could not have dimension 1 for any f in $\mathbb Z[x,y]$ given that I know the dimension of $\mathbb Z[x,y]$ is 3. Is there another way to see this ? $\endgroup$ – Paul Slevin May 22 '12 at 2:01
  • $\begingroup$ If you know that every maximal ideal has height 3 in your ring, the problem is solved. $\endgroup$ – wxu May 22 '12 at 2:07

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