Exercise 11, § 2

Let $A$ be a ring, $M$ an $A$-module and $N$ a submodule of $M$. The set of $a \in A$ such that the homothety of ratio $a$ on $M/N$ is almost nilpotent is called the radical of $N$ in $M$ and denoted by $r_M(N)$. Show that $r_M(N)$ is an ideal of $A$ and that, for all $p \in \operatorname{Ass}_f(M/N)$, $r_M(N) \subset p$. If $N_1, N_2$ are two submodules of $M$, then $r_M(N_1 \cap N_2) = r_M(N_1) \cap r_M(N_2)$; if $a$ is an ideal of $A$, $r_M(aN) \supset r(a) \cap r_M(N)$ and $r_A(a) = r(a)$.