## Ideal Quotient (Colon Ideal) is an Ideal

## Problem 203

Let $R$ be a commutative ring. Let $S$ be a subset of $R$ and let $I$ be an ideal of $I$.

We define the subset

\[(I:S):=\{ a \in R \mid aS\subset I\}.\]
Prove that $(I:S)$ is an ideal of $R$. This ideal is called the * ideal quotient*, or

*.*

**colon ideal**