Exercise 12, § 1

Let G be a module and E, F two submodules such that $E \subset F$.

(a) If F is a direct factor of G, $F/E$ is a direct factor of $G/E$. If also E is a direct factor of F, E is a direct factor of G.

(b) If E is a direct factor of G, then E is a direct factor of F. If also $F/E$ is a direct factor of $G/E$, then F is a direct factor in G.

(c) Give an example of two submodules M, N of the $\mathbf{Z}$-module $E = \mathbf{Z}^2$ such that M and N are direct factors of E but $M + N$ is not a direct factor of E.

(d) Let $p$ be a prime number and $U_p$ the submodule of the $\mathbf{Z}$-module $\mathbf{Q}/\mathbf{Z}$ consisting of the classes mod $\mathbf{Z}$ of rational numbers of the form $k/p^n$ ($k \in \mathbf{Z}$, $n \in \mathbf{N}$). Let E be the product $\mathbf{Z}$-module $M \times N$, where M and N are both isomorphic to $U_p$, M and N being canonically identified with submodules of E. Consider the endomorphism $u : (x, y) \mapsto (x, y + px)$ of E; show that $u$ is bijective. If $M' = u(M)$, show that M and $M'$ are both direct factors of E but that $M \cap M'$ is not a direct factor of E (use (b), showing that the $\mathbf{Z}$-module $U_p$ has no direct factor other than itself and $\{0\}$).