Exercise 16, § 4

Let M be a module over a commutative ring A. A submodule N of M is called characteristic if $u(N) \subset N$ for every endomorphism $u$ of M. For all $a \in A$, the submodule $M(\alpha)$ of elements annihilated by a, and the submodule $\alpha M$, are characteristic. a) If N is a characteristic submodule of M, and P and Q are two complementary submodules of M, show that N is the direct sum of $N \cap P$ and $N \cap Q$ (consider the projections from M onto P and Q). b) Let M be a module of finite length over a principal ideal domain A ; let $Aa_i$ be the invariant factors of M, arranged in decreasing order ($1 \leq i \leq n$) ; let $(M_i)_{1 \leq i \leq n}$ be a sequence of submodules of M such that M is the direct sum of the M, and each $M_i$ is isomorphic to $A/A\alpha_i$. Let N be a characteristic submodule of M and let $A\beta_i$ be the annihilator of $N \cap M_i$ ($1 \leq i \leq n$) ; if $a_i = \beta_i \gamma_i$, show that $\beta_i$ divides $\beta_{i+1}$, and $\gamma_i$ divides $\gamma_{i+1}$ for all $1 \leq i \leq n - 1$ (notice that, for $1 \leq i \leq n - 1$, there exists an endomorphism of M which maps $M_{i+1}$ onto $M_i$, and an endomorphism of M which maps M, onto a submodule of $M_{i+1}$). c) Conversely, let N be a submodule of M which is a direct sum of submodules $N_i \subset M_i$; suppose the annihilators $A\beta_i$ of the N, ($1 \leq i \leq n$) satisfy the above conditions. Then show that N is a characteristic submodule of M. Deduce that if two characteristic submodules of M have the same invariant factors then they are equal.