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)}.

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