Exercise 1, § 2

(a) If A is a ring with no divisors of zero, show that every element $\neq 0$ of a projective A-module is free.

(b) Give an example of a projective A-module whose annihilator is non-zero (cf. I, § 8, no. 11).

(c) Show that the $\mathbf{Z}$-module $\mathbf{Q}$ is not a projective $\mathbf{Z}$-module. (See Commutative Algebra, II, § 5, Exercise 11 for an example of a finitely generated projective module over an integral domain, which is not free.)