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Problem: Let $R=\mathbb{Z}[t]$ and $I$ an ideal of $R$. Then $R/I$ is finitely generated as a $\mathbb{Z}$-module if and only if $I$ contains a monic polynomial.

Suppose $I$ contains a monic polynomial $g$, then for each $f\in R$ there are $h,r\in R$ such that $$f=gh+r,\,\deg(r)<\deg(g).$$ But $gh\in I$, so $f=r\bmod{I}$ and the degree of $r$ is bounded. If $\deg(g)=n$, then $R/I$ is generated by, at most, $\mathbb{Z}+t\mathbb{Z}+\dots+t^{n-1}\mathbb{Z}$, hence $R/I$ is finitely generated as a $\mathbb Z$-module. I think this first part is correct, but I'm not really sure how to approach the other one.

Any hints or suggestions? Many thanks in advance!

user26857
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defacto
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1 Answers1

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Let $f_1,\dots,f_n$ be a system of generators, and $m=\max_i\deg f_i$. Now write $t^{m+1}$ as a linear combination of $f_i$s. What you get?

user26857
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