Exercise 12, § 5

Let A be a ring, and let P be a finitely generated A-module. Prove that the following properties are equivalent:

(i) The A-module P is projective and generating.

(ii) The A-modules $P^{(\mathbf{N})}$ and $A^{(\mathbf{N})}_s$ are isomorphic.

(To establish the implication (i)$\Rightarrow$(ii), use induction to construct strictly increasing sequences $(n_k)$ and $(m_k)$ of integers and surjective homomorphisms of A-modules $f_k: A^{n_k}_s\rightarrow P^{m_k}$ and $g_k: P^{m_{k+1}}\rightarrow A^{n_k}_s$ such that $g_{k-1}\circ f_k$ and $f_k\circ g_k$ are the canonical projections.)