Exercise 13, § 2

Let A be a local integral domain that is not a field (for example $\mathbf{Z}_{(p)}$, cf. VIII, p. 26, Example 5), and let $a$ be a nonzero noninvertible element of A. Denote the A-module $A^{(\mathbf{N})}$ by M and its canonical basis by $(e_n)_{n\in\mathbf{N}}$. Prove that the sequence $(f_n)_{n\in\mathbf{N}}$, where $f_{2m}=e_{2m}$ and $f_{2m+1}=e_{2m+1}+ae_{2m+2}$ for every $m\geqslant 0$, is a basis of M over A. Prove that the submodule $N =\sum_mA(e_{2m}+ae_{2m+1})$ of M is a direct factor but that there is no subset J of $\mathbf{N}$ such that$\sum^{\geqslant 0}_{n\in J}Af_j$ is supplementary to $\mathbf{N}$. Deduce that in Corollary 5 of VIII, p. 35, we cannot leave out the assumption that I is finite.