Exercise 32, § 1

Let $A$ be an integral domain $(a_i)_{1 \leq i \leq n}$ a finite family of elements of $A$, $a$ the ideal $\sum_i A a_i$ and $R$ the submodule of $A''$ generated by the element $(a_1, \ldots, a_n)$. Show that, for the torsion submodule of the $A$-module $M = A^n/R$ to be a direct factor of $M$, it is necessary and sufficient that