Property of colon ideal: $I: (J+K)=(I:J)\cap(I:K)$

Question

Let $I,J,K$ are ideals in a commutative ring with unity $R$ Then $I:(J+K)=(I:J)\cap(I:K)$

My solution is if $a\in I: (J+K)$ then $a(j+k)\in I$. For all $j+k\in J+K$ $aj+ak\in I$ but $aj \in I$ and $ak \in I$ hence $aj \cap ak \in I$ so I don't know how I can prove $a$ is in another set. I want some help here.

Answer 1

I am assuming that you want to prove $$I:(J+K)=(I:J)\cap (I:K).$$If $a\in I:(J+K)$ then $aj+ak\in I$ for all $j\in J$ and $k\in K$ as you already noted. So taking $k=0$ in particular, we can say that $aj\in I$ for all $j\in J$ and taking $j=0$, we can say $ak\in I$ for all $k\in K$ that is, $a\in (I:J))$ and $a\in (I:K)$.