## A basic property of ideal quotients in a quotient ring

Let $R$ be a ring and let $I$, $J$, and $K$ be ideals in $R$ such that $K \subseteq I$. If $\overline{I}$ and $\overline{J}$ denote the images of $I$ and $J$ in $R/K$ via the natural projection, prove that $\overline{(I:J)} = (\overline{I}:\overline{J})$ where $\overline{(I:J)}$ is the image of the ideal quotient $(I:J)$.

$(\subseteq)$ Suppose $r + K \in \overline{(I:J)}$. Then $r \in (I:J)$, and thus $rJ \subseteq I$. Then $rJ + K \subseteq I + K$, and so $(r+K)(J + K) \subseteq I+K$. So $r+K \in (\overline{I}:\overline{J})$.

$(\supseteq)$ Suppose conversely that $r+K \in (\overline{I}:\overline{J})$. Then $(r+K)(J+K) \subseteq I+K$, and thus $rJ+K \subseteq I+K$. Since $K \subseteq I$, $rJ \subseteq I$. So $r \in (I:J)$, and $r+K \in \overline{(I:J)}$.