For a PID $R,$ what is the rank of the submodule given by $S=\{(x_1,\ldots,x_n) \in R^n: x_1 +\cdots +x_n=0\}$?

Question

Let $R$ be a PID and $R^n$ be the free $R$-module of rank $n.$ Now consider the $R$-submodule of $R^n$ given by $$S=\{(x_1,\ldots,x_n) \in R^n: x_1 +\cdots +x_n=0\}.$$ We know that $S$ is a free $R$-module of rank less than or equal to $n.$ In particular when $R$ is a field then $S$ is of rank $n-1.$ In general can we prove that $\text{rank}(S)=n-1$ ?