Exercise 15, § 1

Let c be an infinite cardinal.

(a) Let I be a set and R $\{a, b\}$ a relation between elements of I such that for all $\alpha \in I$ the set of $\beta \in I$ for which $R \{ \alpha, \beta \}$ holds has cardinal $\leq c$. Show that there exists a well-ordered set L and an increasing family $(I_\lambda)_{\lambda \in L}$ of subsets of I such that: (1) there exists a $\lambda \in L$ such that $I_\lambda = \varnothing$; (2) I is the union of the $I_\lambda$ for $\lambda \in L$; (3) if $\lambda \in L$ is such that the set of $\mu < \lambda$ has a greatest element $\lambda'$, $\mathrm{Card}(I_\lambda - I_{\lambda'}) \leq c$; (4) if $\lambda \in L$ is the least upper bound of the set of $\mu < \lambda$, $I_\lambda$ is the union of the $I_\mu$ for $\mu < \lambda$; (5) if $\alpha \in I_\lambda$ and $R \{ \alpha, \beta \}$ holds, then $\beta \in I_\lambda$. (Note first that for all $\alpha \in I$ the set of $\beta \in I$ for which there exists an integer n and a sequence $(\gamma_j)_{1 \leq j \leq n}$ of elements of I such that $\gamma_1 = \alpha$, $\gamma_n = \beta$ and $R \{ \gamma_j, \gamma_{j+1} \}$ holds for $1 \leq j \leq n-1$, has cardinal $\leq c$. Then construct the $I_\lambda$ by transfinite induction.)

(b) Let E be an A-module the direct sum of a family $(M_\alpha)_{\alpha \in I}$ of submodules such that $\gamma(M_\alpha) \leq c$ for all $\alpha \in I$ (cf. Exercise 6). Let f be an endomorphism of E. Show that there exists a well-ordered set L and an increasing family $(I_\lambda)_{\lambda \in L}$ of subsets of I satisfying properties (1), (2), (3) and (4) of (a) and such

LINEAR ALGEBRA

moreover that if we write $E_\lambda = \bigoplus_{\alpha \in I_\lambda} M_\alpha$, then $f(E_\lambda) \subset E_\lambda$. (Apply (a) to the relation R$\alpha, \beta$: "there exists $x \in M_\alpha$ such that the component of $f(x)$ in $M_\beta$ is $\neq 0$".) For all $\lambda \in L$ such that the set of $\mu < \lambda$ has a greatest element $\lambda'$, write $F_\lambda = \bigoplus_{\alpha \in I_\lambda - I_{\lambda'}} M_\alpha$. Show that E is the direct sum of the $F_\lambda$ (apply Exercise 14).

(c) With the hypotheses and notation of (b), suppose further that $f$ is a projector and write $P = f(E)$. Let $P_\lambda = P \cap E_\lambda$; show that if $\lambda \in L$ is such that the set of $\mu < \lambda$ has a greatest element $\lambda'$, $P_{\lambda'}$ is a direct factor of $P_\lambda$ (cf. Exercise 12), that a supplementary submodule $P'_\lambda$ of $P_{\lambda'}$ in $P_\lambda$ is isomorphic to a direct factor of $F_\lambda$ and that $\gamma(P'_\lambda) \leq c$. Show that $P$ is the direct sum of the $P'_\lambda$ (apply Exercise 14).

(d) Deduce from (c) that if an A-module E is the direct sum of a family $(M_\alpha)_{\alpha \in I}$ of submodules such that $\gamma(M_\alpha) \leq c$, then every direct factor P of E is also the direct sum of a family $(N_\lambda)_{\lambda \in L}$ of submodules such that $\gamma(N_\lambda) \leq c$.