Exterior Algebras and Fraction Fields (11.5.8c)

Dummit and Foote Abstract Algebra, section 11.5, exercise 8(c):

(c) Give an example of an integral domain $R$ with fraction field $F$ and ideal $I⊆F$ considered as an $R$-module such that $\bigwedge^n I \neq 0$ for all $n$.

Proof: Let $R = \mathbb{Z}[x_1,...]$ and $I = (x_1,...)$. It suffices to find an alternating $n$-multilinear map $φ_n : I × ... × I → \mathbb{Z}$ ($n$ factors) such that $φ_n(x_1,...,x_n) = 1$ for all $n$. To that end, define $φ_n$ as follows,$$φ_n: I × ... × I → \mathbb{Z}$$$$φ_n(∑a_{1,i}x_i,...,∑a_{n,i}x_i)=\text{det }(a_{ij}')_{1≤i,j≤n}$$where $a'_{ij}$ is the constant term of $a_{ij}$. Note that $∑a_{i}x_i=∑b_{i}x_i$ implies that $a_i'=b_i'$ for all $i$, so that $(a_{ij}')_{1≤i,j≤n}$ is uniquely determined and $φ_n$ is well defined. This map is multilinear alternating on components of $I × ... × I$ just as $\text{det}$ is multilinear alternating on matrix rows. Here $(x_1,...,x_n)$ represents the identity matrix, and as such $φ_n(x_1,...,x_n) = 1$.$\square$