# Noetherian ring

Let $R$ be a noetherian ring. By the Hilbert Basis Theorem the polinomial ring $R[x_1, ... , x_n]$ is also a noetherian ring. What can we say about the number of generators of an ideal $I$ of $R[x_1, ... , x_n]$? (We can suppose that every ideal in $R$ is principal)

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Nothing in this generality. There is a whole book about special cases: amazon.de/Ideals-Reality-Projective-Mathematics-ebook/dp/… –  darij grinberg Jan 16 '12 at 19:29
this paper on a special case (ideals of reduced affine curves) may be of interest. websupport1.citytech.cuny.edu/faculty/hschoutens/PDF/… –  roy smith Jan 17 '12 at 20:38

Nothing. Assume $R=k$, a field, for specificity. Then $k[x_1]$ is a principal ideal domain, as you know, but $k[x_1,x_2]$ has ideals with unbounded number of generators. Specifically, $(x_1,x_2)^n$ is minimally generated by $n+1$ elements for all $n$. One can get higher rates of growth by adding more variables.

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