Let $R$ be a Noetherian ring. By the Hilbert Basis Theorem the polynomial ring $R[x\_1, \ldots , x\_n]$ is also a Noetherian ring. What can we say about the number of generators of an ideal $I$ of $R[x\_1, \ldots , x\_n]$? (We can suppose that every ideal in $R$ is principal)



Number of generators of an ideal in a polynomial ring over a Noetherian ring
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.


