Exercise 17, § 2

(*) Let $A$ be a ring (not necessarily commutative) and $\mathscr{F}$ a topologizing set (Exercise 16) of left ideals of $A$. (a) $A$ (left) $A$-module $M$ is called 9-negligible if the annihilator of every element of $M$ belongs to $\mathscr{G}$ If $M$ is $\mathscr{F}$-negligible, so is every submodule and every quotient module of $M$. If $M$ and $N$ are two $\mathscr{F}$ negligible $A$-modules, $M \oplus N$ is 9-negligible. For $A_s/m$ to be 9-negligible, it is necessary and sufficient that $m \in \mathscr{G}$ If $(0) \in \mathscr{G}$ every $A$-module is 9-negligible; if consists only of the single left ideal $A_s$, every $\mathscr{F}$-negligible $A$-module is reduced to 0. (b) For every $A$-module $M$ there exists a greatest 9-negligible submodule of $M$ which is denoted by $\mathscr{F}M$. For every $A$-module homomorphism $u : M \to N$, $u(\mathscr{F}M) \subset \mathscr{F}N$; let $\mathscr{F}u$ be the mapping from $PM$ to $\mathscr{F}N$ whose graph coincides with that of $u|_{\mathscr{F}M}$. If

is an exact sequence of $A$-modules, the sequence is exact. Show that $\mathscr{F}A_s$ is a two-sided ideal of $A$. (c) Let $m, n$ be two elements of $\mathscr{F}$ such that $m \subset n$; for every left $A$-module $M$ the canonical injection $j : m \to n$ defines a commutative group homomorphism

(*) Exercises 17 to 25 (unedited) were communicated to us by P. Gabriel.

If $\mathscr{F}$ is ordered by the relation $\supseteq$, the $u_{m,n}$ define a direct system of commutative groups, whose direct limit will be denoted by $M_{(\mathscr{F})}$. The mappings $u_{m,A_s}$ form a direct system, whose direct limit is a mapping (called canonical) from $M = \mathrm{Hom}_A(A_s, M)$ to $M_{(\mathscr{F})}$. Similarly, if $u : M \to N$ is a left $A$-module homomorphism, the $v_m = \mathrm{Hom}_A(1, u) : \mathrm{Hom}_A(m, M) \to \mathrm{Hom}_A(m, N)$ form a direct system, whose direct limit is a commutative group homomorphism

If $0 \to M \xrightarrow{v} N \xrightarrow{w} P$ is an exact sequence of $A$-modules, the sequence

is exact.