Proving the equality $eR\cap (I+J)=(eR\cap I)+(eR\cap J)$

Question

Let $I, J$ be left-ideals of an unitary ring $R$, and let $e\in R$ be idempotent. Then, it must be shown that $$eR\cap (I+J)=(eR\cap I)+(eR\cap J).$$

I've tried to show that $eR\cap (I+J)\subseteq (eR\cap I)+(eR\cap J) $ and $eR\cap (I+J)\supseteq (eR\cap I)+(eR\cap J).$ However, I have some troubles at understanding how does an element $x\in eR\cap (I+J) $ look like (the same goes for a $y\in (eR\cap I)+(eR\cap J) .$ (I think this is my main trouble)

Thanks in advance!

Answer 1

However, I have some troubles at understanding how does an element $x\in eR\cap (I+J) $ look like

Nothing surprising, if that's what you are afraid of. $x=er=i+j$ for some $r\in R, i\in I,j\in J$.

Then $i+j=e(i+j)=ei+ej\in (eR\cap I) + (eR \cap J)$. It's that straightforward.

The other containment is even easier: if $i\in eR\cap I$ and $j\in eR\cap J$ then $i+j\in eR$ since $eR$ is additively closed, and it is trivially in $I+J$.