I'm looking for a reference for the following fact.

Suppose $A$ is a finitely generated associative commutative unital algebra over an algebraically closed field of characteristic zero. Let $S^n(A)$ be the $n$-th symmetric power of $A$. That is, $S^n(A)$ is the subalgebra of $A^{\otimes n}$ fixed by the natural action of $S_n$ on this tensor product by permuting the factors. I would like to know a reference for the fact that $S^n(A)$ is generated by elements of the form

$(a \otimes 1 \otimes \cdots \otimes 1) + (1 \otimes a \otimes 1 \otimes \cdots \otimes 1) + \cdots + (1 \otimes \cdots \otimes 1 \otimes a)$

for $a$ in $A$.

Note that if $A = \mathbb{C}[t]$, then $S^n(A)$ is the ring of symmetric polynomials in $n$ variables and then the result follows from the fact that the power sum polynomials generate this ring.

I call the above a "fact" since I've seen it used in papers (without proof -- and I've contacted the authors and they don't know of a reference) and a colleague of mine says he could write out a proof. However, I can't help but think that there must be a reference for this in some text. But I'm unable to find one...

Alternatively, if someone knows of a very short proof, that would be the next best thing to a reference. I want to use this fact in a paper I'm writing and I'd like to avoid having to digress from the main thread of the exposition by including a long proof of this fact.