Symmetric powers of complete modules over a two-dimensional regular local ring

Let $(R,m)$ be a two-dimensional regular local ring with infinite residue field. For a finitely generated, torsion-free $R$-module $A$, write $A_{n}$ for the $n$th symmetric power of $A$, mod torsion. We study the modules $A_{n}$, $n \geq 1$, when $A$ is complete (i.e., integrally closed). In particular, we show that $B\cdot A = A_{2}$, for any minimal reduction $B \subseteq A$ and that the ring $\oplus _{n \geq 1} A_{n}$ is Cohen-Macaulay.