Exercise 23, § 2

Let M be an A-module, the direct sum of a family (finite or otherwise) $(M_\lambda)_{\lambda \in L}$ of submodules such that for all $\lambda \in L$ the non-invertible elements of $\operatorname{End}(M_\lambda)$ form a two-sided ideal of this ring (which implies that $M_\lambda$ is indecomposable).

(a) Let $f, g$ be two endomorphisms of $M$ such that $f + g = 1_M$. Show that for every finite sequence $(\lambda_k)_{1 \leq k \leq s}$ of distinct indices in $L$ there exists a family of submodules $N_k$ $(1 \leq k \leq s)$ of $M$ such that for every $k$ one of the endomorphisms $f, g$ restricted to $M_{\lambda_k}$ is an isomorphism of this submodule onto $N_k$ and that $M$ is the direct sum of the $N_k$ $(1 \leq k \leq s)$ and the $M_\lambda$ for $\lambda$ distinct from $\lambda_k$. (Reduce it to the case $s = 1$; if $\pi_\lambda$ is the canonical projection of $M$ onto $M_\lambda$ corresponding to the decomposition as a direct sum of $M_\lambda$ and $\bigoplus_{\mu \neq \lambda} M_\mu$, note that either $\pi_\lambda \circ f$ or $\pi_\lambda \circ g$, restricted to $M_\lambda$, is an automorphism of $M_\lambda$.)

(b) Let $f$ be an idempotent endomorphism of $M$. Show that there exists at least one index $\lambda \in L$ such that the restriction of $f$ to $M_\lambda$ is an isomorphism of $M_\lambda$ onto $f(M_\lambda)$; moreover $f(M_\lambda)$ is a direct factor of $M$. (If $g = 1_M - f$, note that there exists a finite sequence of indices $(\lambda_k)_{1 \leq k \leq s}$ such that the intersection of $\operatorname{Ker}(g)$ and the direct sum of the $M_{\lambda_k}$ is $\neq \{0\}$ and use (a).)

(c) Deduce from (b) that every indecomposable direct factor of $M$ is isomorphic to one of the $M_\lambda$.

(d) Let $(N_\kappa)_{\kappa \in K}$ be a family of indecomposable submodules of $M$ of which $M$ is the direct sum; every $N_\kappa$ is therefore isomorphic to an $M_\lambda$ and vice versa, by virtue of (c). Let $T$ be the set of classes of indecomposable submodules of $M$ (under the relation of isomorphism) such that an $M_\lambda$ (or an $N_\kappa$) belongs to one of these classes. For all $t \in T$, let $R(t)$ (resp. $S(t)$) be the set of $\lambda \in L$ (resp. $\kappa \in K$) such that $M_\lambda \in t$ (resp. $N_\kappa \in t$). Show that, for all $t \in T$, $\operatorname{Card}(R(t)) = \operatorname{Card}(S(t))$. (In the notation of (a), let $J(\kappa)$ be the set $\lambda \in L$ such that the restriction of $\pi_\lambda$ to $N_\kappa$ is an isomorphism of $N_\kappa$ onto $M_\lambda$; show that $J(\kappa)$ is finite and that the $J(\kappa)$ cover $R(t)$ when $\kappa$ runs through $S(t)$. Deduce that $\operatorname{Card}(S(t)) \leq \operatorname{Card}(R(t))$ when $R(t)$ is infinite. When $R(t)$ is finite, show with the aid of (a) that $M$ is the direct sum of the $M_\lambda$ for $\lambda \notin R(t)$ and a subfamily of $(N_\kappa)_{\kappa \in S(t)}$ of cardinal equal to that of $R(t)$.)

(e) Deduce from (d) and Exercises 22 and 21 (d) that if $(E_\lambda)_{\lambda \in L}$ and $(E'_\kappa)_{\kappa \in K}$ are two reduced irreducible decompositions of $\{0\}$ in a module $F$, $L$ and $K$ are equipotent.