Suppose $R$ is a commmutative ring and $I,J,K$ are ideals. In Atiyah-Macdonald's Commutative Algebra, it is mentioned that if $J\subseteq I$ or $K\subseteq I$, then $I\cap(J+K)=I\cap J+I\cap K$, but I can't figure out why it's true.
So WLOG suppose $J\subseteq I$. Let $x\in I\cap(J+K)$. We wish to show that $x\in I\cap (J+K)$. We have $x=i$ and $x=j+k$ for some $i\in I$ and $j\in J$ and $k\in K$. How should I proceed? Is this even the right way to think about it?
I don't have enough reputation to comment so I'll write up a solution. WLOG assume that $J\subset I$. In order to prove $$ I\cap(J+K)=I\cap J+I\cap K $$ we must show they are subsets of each other. I'll show the inclusion that $$ I\cap(J+K)\subset I\cap J+I\cap K $$ So let $x\in I\cap(J+K)$, then $x=i\in I$ and $x=j+k\in(J+K)$. We know $J\subset I$ thus $j\in I$. Also $$ i-j=k $$ since $I$ is an ideal, $i-j=k\in I$. Thus $k\in I$. And now since we assumed $x=j+k$ we have shown it is of the form something in I and J plus something in I and K. Thus $$ x\in I\cap J+I\cap K $$ I'll trust the other inclusion to you.