Let $V$ be an $R$-module. Traditionally the symmetric algebra $S(V)$ is defined as a quotient of the tensor algebra $T(V)$, by the ideal generated by all $a\otimes b-b\otimes a$.
Can $S^n(V)$ also be viewed as a submodule of $T(V)$?
This is clearly true if $n!$ is invertible in $R$, by mapping $a_1\cdots a_n$ to the symmetrized tensor $1/n!\sum a_{\pi(1)}\otimes\cdots\otimes a_{\pi(n)}$. However, I'm interested in a universal construction, valid over all rings.
In particular, without the $1/n!$, the above map would not necessarily be injective: it maps $a^n$ to $n! a^n$ which could be $0$ if the characteristic of $R$ divides $n!$.
Note the related question MO69940 asking about splitting the natural map $T(V)\to S(V)$. I ask for something weaker, namely just an injective module map that is natural in $V$.