Ideal quotient
= Ideal quotient
{title2=$(I:J)=\{r:rJ\subseteq I\}$}
For <ideals> $I,J$ in a commutative <ring>, the ideal quotient is $(I:J)=\{r\in R:rJ\subseteq I\}$. Closure under subtraction and multiplication by arbitrary ring elements proves it is an <ideal>. If $I\subseteq J=(a)$, then $I=a(I:J)$: write $x=ar\in I$ and observe that $rJ=xR\subseteq I$. Consequently, if also $(I:J)=(b)$, then $I=(ab)$ is a <principal ideal>.