# Ideals of every “dimension” in the ring Z[x]

Let Z[x] be the ring of polynomials with integer coefficients.

I need help proving that given any natural number m there exists a collection of m polynomials such that the ideal generated by these is not generated by any collection of m-1 polynomials. I.e, there are ideals of every dimension in Z[x].

Where by "dimension" of an ideal J, I mean the least number of elements needed to generate J.

• That's an unusual definition of dimension. Can you prove that if $M$ is a maximal ideal of the commutative ring $R$, $k=R/M$ is the quotient field, and $I$ is any ideal of $R$, then $I$ requires at least $m$ generators where $m=\dim_k I/MI$? – Angina Seng Feb 3 '19 at 5:14

Peeking at this answer , I'm going to venture the following guess: $$(2^m,2^{m-1}x+2^{m-1},\dots,2x^{m-1}+2)$$.
On the other hand, this answer may be more to the point. Apparently we could do: $$(p^n,p^{n-1}x,\dots,px^{n-1},x^n)$$, to get an ideal that can't be generated by fewer than $$n+1$$ elements.