Proposition 25

Let $A$ be an integral domain and $(E_i)$ a family of $A$-modules; then

Let $(x_i)$ be an element of $\bigoplus_i E_i$ such that $x_i \in T(E_i)$ for all $i$; then each of the $x_i$ is annihilated by an element $\alpha_i \neq 0$ of $A$ and it may be assumed that $\alpha_i = 1$ when $x_i = 0$; as the family $(x_i)$ has finite support, the element $\alpha = \prod_i \alpha_i$ of $A$ is defined and $\neq 0$; it obviously annihilates $\bigoplus_i x_i$ and hence $\bigoplus_i T(E_i) \subset T\left(\bigoplus_i E_i\right)$; the converse is immediate.

If $E$ and $F$ are two $A$-modules, clearly $T(E \otimes_A F)$ contains the canonical images of $T(E) \otimes_A F$ and $E \otimes_A T(F)$; but examples can be given of torsion-free $A$-modules $E, F$ such that $T(E \otimes_A F) \neq 0$ (Exercise 31).

Note that an infinite product of torsion modules is not necessarily a torsion module; for example, in the $\mathbf{Z}$-module $\prod_{n=1}^\infty (\mathbf{Z}/p^n\mathbf{Z})$ ($p$ an integer > 1), the element all of whose coordinates are 1 is free.