Exercise 6, § 4

Let A be a commutative ring and let E be an A-module. Let $\mathscr{E}$ denote the ring $\mathrm{End}_\mathbf{Z}(E)$ of endomorphisms of the abelian group E (without operators); for every subring $B \subset \mathscr{E}$, let $B'$ denote the centraliser of B in $\mathscr{E}$ (the subring of $\mathscr{E}$ consisting of elements which commute with every element of B). a) Let $A_0$ be the subring of $\mathscr{E}$ consisting of the homotheties $x \mapsto \lambda x$ ($\lambda \in A$); if a is the annihilator of E, then $A_0$ is known to be isomorphic to $A/a$, and its centraliser $A_0'$ in $\mathscr{E}$ is the ring $\mathrm{End}_A(E)$ of endomorphisms of E as an A-module. Show that the centraliser $A_0''$ of $A_0'$ is commutative, and that its centraliser $A_0'''$ is equal to $A_0'$ (notice that $A_0 \subset A_0'$). b) If the submodule F is a direct factor of E, show that $u(F) \subset F$ for all $u \in A_0'$ (consider the projection of E onto F obtained from the decomposition of E as a direct sum of F and another submodule). c) Suppose that E is the direct sum of a family $(F_i)$ of cyclic submodules. If $u \in A_0'$ show that, for each index i, there exists an element $a_i \in A$ such that $u(x) = a_i x$ for all $x \in F_i$ (use b)). d) Suppose E is the direct sum of a sequence $(E_n)$ (finite or infinite) of cyclic submodules such that, if $b_n$ denotes the annihilator of $E_n$, then the sequence $(b_n)$ is increasing. Show that $A_0' = A_0$ (use c), and the fact that for each index $i > 1$ there exists an endomorphism $v_i \in A_0'$ which maps $E_i$ into $E_i$. Apply this to the case where A is a principal ideal domain and E is a finitely generated A-module.