Exercise 8, § 5

Let A be an integral domain and K its field of fractions; for every A-module M, let $\tau(M)$ denote its torsion module (II, § 7, no. 10).

(a) Show that for the algebra $T(M)$ to have no divisors of zero, it is necessary and sufficient that the A-module $T(M)$ be torsion-free (observe that $T(M)_K$ is isomorphic to $T(M_K)$ and use Exercise 5).

(b) Let $u : M \to M'$ be an injective A-module homomorphism. Show that $\operatorname{Ker}(T(u)) \subset \tau(T(M))$; if $M'$ is torsion-free, then $\operatorname{Ker}(T(u)) = \tau(T(M))$ (cf. II, § 7, no. 10, Proposition 27).

(c) Give an example of a torsion-free A-module M such that $\tau(T(M))$ is non-zero (II, § 7, Exercise 31).