Exercise 28, § 9

Let A be an absolutely flat ring (Exercise 27) and $n$ an integer.

a) Prove that every finitely generated submodule of $A^n$ admits a supplementary submodule. (Reason by induction on $n$, by showing that $M\cap A^{n-1}$ is finitely generated and choosing supplementary submodules of $M\cap A^{n-1}$ in $A^{n-1}$ and of the image of M in $Ae_n.$)

b) Deduce that the ring $\mathbf{M}_n(A)$ is absolutely flat (cf. VIII, p. 114, Example 2).