## Submodule Consists of Elements Annihilated by Some Power of an Ideal

## Problem 417

Let $R$ be a ring with $1$ and let $M$ be an $R$-module. Let $I$ be an ideal of $R$.

Let $M’$ be the subset of elements $a$ of $M$ that are annihilated by some power $I^k$ of the ideal $I$, where the power $k$ may depend on $a$.

Prove that $M’$ is a submodule of $M$.