Exercise 22, § 3

(*) Let $A$ be a ring, commutative or not; all the $A$-modules to be considered are left $A$-modules. Let $P$ be a property such that: (α) if $f : M \to N$ is an injective $A$-module homomorphism and $N$ has property $P$, then $M$ has property $P$; (β) the direct sum of two $A$-modules with property $P$ has property $P$.

(a) Let $M$ be an $A$-module. Show that the submodules $M'$ of $M$ such that $M/M'$ has property $P$ form a fundamental system of neighbourhoods of 0 for a linear topology $\mathscr{T}_P(M)$ on $M$. Show that $\mathscr{T}_P(A_s)$ is compatible with the ring structure on $A$ and that $M$ with the topology $\mathscr{T}_P(M)$ is a topological module over $A$ with $\mathscr{T}_P(A_s)$. Every $A$-module homomorphism $f : M \to N$ is continuous with the topologies $\mathscr{T}_P(M)$ and $\mathscr{T}_P(N)$.

(b) Suppose that the following condition holds: (γ) if $N$ is a submodule of $M$ with property $P$ and, for every submodule $L \neq 0$ of $M$, $N \cap L \neq 0$, then $M$ has property $P$. Show that with these conditions, if $F$ is a submodule of an $A$-module $E$, the topology $\mathscr{T}_P(F)$ is induced by $\mathscr{T}_P(E)$. (Let $F'$ be a submodule of $F$ such that $F/F'$ has property $P$; consider a maximal element $G$ among the submodules of $E$ such that $G \cap F = F'$ and show that $E/G$ has property $P$.)