Exercise 10, § 7

Let A be a primitive ring with nonzero socle $\mathfrak{s}$, and let M be an A-module. Then M is faithful and isotypical if and only if we have $\mathfrak{s}M = M$ (use Proposition 4 of VIII, p. 50). Deduce that if $0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$ is an exact sequence of A-modules and $M'$ and $M''$ are faithful and isotypical, then the same holds for M.