Exercise 7, § 2

Let M be a module over a principal ideal domain A; a submodule N of M is said to be pure if, for all $a \in A$, we have $N \cap (\alpha M) = \alpha N$. a) The submodule N of M is pure if and only if, for all $a \in A$ and every element $1 \in M/N$ whose annihilator is $Aa$, there exists an element $x \in \dot{x}$ whose annihilator is $Aa$. b) Show that the torsion submodule of M is pure. A family $(x_i)$ of elements of M is pseudofree (Ex. 4) if and only if the sum of the submodules $Ax_i$ is direct and is a pure submodule of M. Show that if P and Q are submodules of M such that $P \cap Q$ and $P + Q$ are pure then P and Q are pure. If Q is a pure submodule of M and $P \supseteq Q$ a submodule of M, then P is pure in M if and only if $P/Q$ is pure in $M/Q$. c) Show that every submodule N of M which is a direct factor of M is pure. The same is true for the union of a filtered family of direct factors of M. Conversely, if N is a pure submodule of M, and $M/N$ is a direct sum of cyclic submodules, show that N is a direct factor of M (use a)). Give examples of direct factors P and Q of M such that $P \cap Q$ (resp. $P + Q$) is not a pure submodule (for the first example, take $M = (\mathbf{Z}/4\mathbf{Z}) \oplus (\mathbf{Z}/2\mathbf{Z})$, and for the second, take $M = \mathbf{Z}^2$). d) Let N be a pure submodule of M, whose annihilator $Aa$ is nonzero. Iff is the natural homomorphism from M onto $M/\alpha M$, show that f restricts to an isomorphism from N onto $f(N)$, and that $f(N)$ is a pure submodule of $M/\alpha M$ (use Bezout's identity). Deduce (with the help of c) and Ex. 4, c)) that $f(N)$ is a direct factor in $M/\alpha M$, and that $N$ is a direct factor in $M$. In particular, if the torsion submodule of a module $M$ has nonzero annihilator (which is the case when it is finitely generated), then it is a direct factor in $M$ (cf. VII, p. 20, Cor. 1 to Th. 2).

e) Let $p$ be a prime number; let $N$ denote the direct sum of the cyclic $\mathbf{Z}$-modules $\mathbf{Z}/(p^n)$ ($n \geq 1$), and let $e_n$ denote the element of $N$ whose coordinates are all zero except for the $n$-th, which is equal to the class of 1 mod $p^n$. Let $M$ be the submodule of the $\mathbf{Z}$-module $N \times \mathbf{Q}$ generated by the elements $(e_n, p^{-n})$. Show that the torsion submodule $P$ of $M$ is not a direct factor in $M$ (notice that every element of $M/P$ is divisible by $p^n$ for all $n$, and that no element of $M$ has this property except for multiples of $(0, 1)$).