If $A$ and $B$ are proper ideals of a ring with no zero divisors, show that $A\cap B \neq \{0\}$

Question

If $A$ and $B$ are proper ideals of a ring with no zero divisors, show that $A\cap B \neq \{0\}$

Ideals: A subset $I$ of a ring $R$ is called an ideal if

1. $I$ is a subring of $R$

2. For all $a\in I,r\in R$ $ar\in I$ and $ra\in I$.

$A$ and $B$ both satisfy the above. How to show that $A\cap B \neq 0$?

Answer 1

Hint:

If $a\in A$ and $b\in B$, where is $ab$?