This question is a generalisation of this question.
Suppose that $k$ is a commutative ring (we could assume more that it is a field), and suppose that $R$ is a commutative $k$-algebra. Given any $\sigma\in\mathfrak{S}_n$, we could view it as a map of $k$-algebras
$$ f_1\otimes\cdots\otimes f_n\mapsto f_{\sigma(1)}\otimes\cdots\otimes f_{\sigma(n)}. $$
Now consider the map $\varphi:R^{\otimes n+1}\to R^{\otimes n}$ defined by
$$ f_1\otimes\cdots\otimes f_n\otimes f_{n+1}\mapsto f_1\otimes\cdots\otimes f_nf_{n+1}, $$
then for any $\sigma\in\mathfrak{S}_n$, could we prove that the preimage of the ideal in $R^{\otimes n}$ generated by
$\{f_1\otimes\cdots\otimes f_n-f_{\sigma(1)}\otimes\cdots\otimes f_{\sigma(n)}\}$
is the ideal generated by
$\{f_1\otimes\cdots\otimes f_n\otimes1-f_{\sigma(1)}\otimes\cdots\otimes1\otimes f_{\sigma(n)}\}$?
