# Polynomial Rings and Finitely Generated Modules

I've been trying to solve the following question:

Show that the polynomial ring $$\mathbb{Z}[x,y,z]$$ is a finitely generated module over its subring generated by the following three elements:$$x+y+z,\ xy+xz+yz,\ xyz.$$

Could you give me a hint? Say $$R$$ is the ring generated by the above three elements. My idea was to show that $$x$$ is integral over $$R$$, i.e. that the subring $$R[x]$$ of $$\mathbb{Z}[x,y,z]$$ is finitely generated as an R-submodule of $$\mathbb{Z}[x,y,z]$$.

Alternatively, I tried finding the monic polynomial in $$R[w]$$ that $$x$$ satisfies, but neither method worked (or maybe I just missed something).

I am quite confused and feel like I'm doing something completely wrong.

• For more context, see this question. Commented Mar 12 at 21:16
• Indeed, show that $\mathbb{Z}[x+y+z, xy+xz+yz, xyz]\subseteq\mathbb{Z}[x,y,z]$ is an integral extension. Hint: $x,y,z$ are all roots of the same monic polynomial.
– Mark
Commented Mar 12 at 21:17

Set $$a = x + y + z$$, $$b = xy + xz + yz$$, and $$c = xyz$$. Then $$(t-x)(t-y)(t-z) = t^3 - at^2 + bt - c \in R[t]$$ so $$x$$, $$y$$, and $$z$$ are all integral over $$R$$. There is a standard property relating integrality to finitely generated modules: when $$B/A$$ is a ring extension and $$b \in B$$, the following properties are equivalent:

(i) $$b$$ is integral over $$A$$,

(ii) $$A[b]$$ is a finitely generated $$A$$-module,

(iii) $$b$$ is contained in a subring $$S$$ of $$B$$ such that $$A \subset S \subset B$$ and $$S$$ is a finitely generated $$A$$-module.

In the proof of this equivalence, it is the direction from (iii) to (i) that is the most technical.

Anyway, this equivalence implies that the sum and product of elements in $$B$$ integral over $$A$$ is integral over $$A$$, so the set of elements in $$B$$ integral over $$A$$ is a subring of $$B$$.

The proof that (i) implies (ii) shows that when $$b$$ and $$b'$$ in $$B$$ are integral over $$A$$, $$A[b,b']$$ is a finitely generated $$A$$-module, and the same is true for the subring of $$B$$ generated over $$A$$ by any finite set of elements in $$B$$ integral over $$A$$.