Is it true that $I+J\subseteq I\cap J$

Question

Let $R$ be a commutative ring and $I,J$ are ideals in $R$. Is it true that

$$I+J\subseteq I\cap J?$$

I know that $I+J =\{x+y|x\in I, y\in J\}$.

Answer 1

In general: $$I\cap J\subseteq I\subseteq I+J.$$ The converse is true if and only if $I=J.$ One direction is obvious. for the other:
Let $I+J\subseteq I\cap J.$
Then $\forall a\in I,\ a=a+0\in I+J \subseteq I\cap J.$ So $\forall a\in I, a\in J.$ Hence $I\subseteq J.$ Similarly $J\subseteq I.$ Therefore, in this case we should have $I=J$.