Why does this ring have rank $k!$?

Let $R$ be any ring free of rank $k$ over $\mathbb{Z}$ having non-zero discriminant. Let $R^{\otimes k} = R \otimes_{\mathbb{Z}} \cdots \otimes_{\mathbb{Z}} R$. Then $R^{\otimes k}$ is a ring of rank $k^k$ in which $\mathbb{Z}$ lies naturally as a subring via $n \rightarrow n(1 \otimes \cdots \otimes 1)$. Denote by $I_R$ the ideal in $R^{\otimes k}$ generated by elements of the form $$x \otimes \cdots \otimes 1 + 1 \otimes x \cdots \otimes 1 + \cdots + 1 \otimes \cdots \otimes x - \text{Tr} (x)$$ for $x \in R$. Let $$J_R = \{ r \in R^{\otimes k} : nr \in I_R \text{ for some } n \in \mathbb{Z} \}.$$ Then the ring $$R^{\otimes k} / J_R$$ has rank $k!$. I have been trying to work out why this ring has rank $k!$, but I am not quite seeing it yet. I would appreciate any explanation. Thank you very much!

PS This is a ring that comes up in Higher composition laws III'' by Manjul Bhargava.

• I'm sorry, I'm not sure what the pattern to that form is. Is it something obvious I'm missing? – jgon Jul 18 '16 at 20:09
• @jgon Each summand's a bunch of $1$s with $\otimes$s in between, except with one of the $1$s replaced by an $x$. – arctic tern Jul 18 '16 at 20:10
• @arctictern oh right, that makes sense then – jgon Jul 18 '16 at 20:15

The aim of the authors is to define for commutative ring extensions a notion similar to the notion of Galois closure of fields. For a commutative ring $B$ and a commutative $B$-algebra $A$ which is locally free of rank $n$, they construct a commutative ring $G(A|B)$ which is called the $S_n$-closure of $A$ over $B$. If $B\subseteq A$ is a field extension of degree $n$ with Galois group $S_n$, it is easily seen that $G(A|B)$ is exactly the Galois closure of $A$ as a field extension of $B$. The authors investigate several properties of the $S_n$-closure, such as for example the functoriality, and the behavior with respect to base change or to general finite products. They also investigate several cases of natural ring extensions, e.g. monogenic or étale ones. Finally, they extend the notion of $S_n$-closure to schemes.