Modular law for ideals in a commutative ring with 1

Question

How do I show that for ideals $A, B,C$ with $A$ containing $B$ of a ring $R$, $A \cap (B+C)$ is contained in $B+(A \cap C)$?

If there is some $a \in A$,$b \in B$,$c \in C$ such that $a=b+c$ then I think I have to use the property $r\in R$ and $a \in A$ implies $ra \in A$. Am I thinking in the right direction?Please help.

Answer 1

As you say, let $a\in A$, $b\in B$ and $c\in C$ with $a=b+c$. We want to show that $a\in B+(A\cap C$). It will suffice to show that $c\in A\cap C$. We already know that $c\in C$. Can you see why $c\in A$? You will need to use the fact that $B\subset A$.