Exercise 23, § 2

Soit M an A-module, direct sum of a family (finite or not) $(M_\lambda)_{\lambda \in L}$ of submodules such that for every $\lambda \in L$, the non-invertible elements of $\mathrm{End}(M_\lambda)$ form a two-sided ideal in this ring (which implies that $M_\lambda$ is indecomposable). (a) Let $f, g$ be two endomorphisms of M such that $f + g = 1_M$. Now prove 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 the $\lambda$ distinct from the $\lambda_k$. (Reduce to the case $s = 1$; if $\pi_\lambda$ is the canonical projection of M onto $M_\lambda$ corresponding to the decomposition into 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. Now prove 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 $\mathrm{Ker}(g)$ 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 conversely, by virtue of c). Let T be the set of classes of indecomposable submodules of M (for the relation of isomorphism) such that an $M_\lambda$ (or an $N_\kappa$) belongs to one of these classes. For every $t \in T$, let $R(t)$ (resp. $S(t)$) be the set of $\lambda \in L$ (resp. $\kappa \in K$) such that $M_\lambda \in t$ (resp. $N_\kappa \in t$). Now prove that for every $t \in T$, one has $\mathrm{Card}(R(t)) = \mathrm{Card}(S(t))$. With the notation of a), let $J(\kappa)$ be the set of $\lambda \in L$ such that the restriction of $\pi_\lambda$ to $N_\kappa$ is an isomorphism of $N_\kappa$ onto $M_\lambda$; prove that $J(\kappa)$ is finite and that the $J(\kappa)$ cover $R(t)$ when $\kappa$ runs through $S(t)$. Deduce that $\mathrm{Card}(S(t)) \leq \mathrm{Card}(R(t))$ when $R(t)$ is infinite. When $R(t)$ is finite, prove with the help of a) that M is the direct sum of the $M_\lambda$ for $\lambda \notin R(t)$ and of 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 indecomposable decompositions of {0} in a module F, L and K are equipotent.