Exercise 18, § 2

Let $A$ be an absolutely flat ring (Exercise 17). (a) Let $P$ be a projective right $A$-module. Show that every finitely generated submodule $E$ of $P$ is a direct factor of $P$. (Reduce it to the case where $P$ is free and finitely generated. Then note that $P/E$ is finitely presented and use Exercises 15.) (b) Show that every projective right $A$-module $P$ is the direct sum of monogenous submodules isomorphic to monogenous right ideals of $A$. (Use Kaplansky’s Theorem (Algebra, Chapter II, § 2, Exercises 3) to reduce the problem to the case where $P$ is generated by a countable family of elements, then use (a).) (c) Give an example of an absolutely flat ring $A$ and a non-projective finitely generated $A$-module (consider a quotient of $A$ by an ideal which is not finitely generated; cf. Algebra, Chapter VIII, § 6, Exercise 15(f) and Commutative Algebra, Chapter 11, § 4, Exercise 17).