$A$ and $B$ are ideals of a ring ${R}$ such that $A\cap B=\{0\}$. Prove that $st=0$ for every $s\in A, t\in B$.

Question

$A$ and $B$ are ideals of a ring ${R}$ such that $A\cap B=\{0\}$. Prove that $st=0$ for every $s\in A, t\in B$.

I have no idea how to solve it. please help.

Answer 1

I think that $A$ and $B$ must be two-side ideals. In that case it's true that $AB\subseteq A\cap B$, so since $st\in AB$, then $st\in A\cap B=\{0\}$, then $st=0$, for every $s\in A$ and $t\in B$.