Ideal quotient

Remarks

$(I : J)$ is an ideal of $R$ and $I \subseteq (I : J)$. Example: in $\Z$, $(12\Z : 4\Z) = 3\Z$; for $r \in \Z$, $4r \in 12\Z$ holds exactly when $3 \mid r$, and then $rj \in 12\Z$ for every multiple $j$ of $4$. Non-example: $1 \notin (12\Z : 4\Z)$, since $1 \cdot 4 = 4 \notin 12\Z$. If $J \subseteq I$, then $(I : J) = R$. The quotient $(0 : J)$ is the annihilator of $J$. Other notations are $I : J$ and $(I :_R J)$; the name colon ideal is also used.