Non-zero Ideal in an Integral Domain is indecomposable

Question

Need to prove that : any non-zero ideal in an integral domain is indecomposable.

Now if $I=A\bigoplus B$, then $A$ and $B$ are subgroups of $I$, if I am not wrong. Does it say anything about having idempotents, then in that case multiplication of the two idempotents will be zero leading to a contradiction that we are in a domain. Please help.

Answer 1

$A\cap B\supseteq AB\neq \{0\}$ If $A$ and $B$ are nonzero.