If $R$ contains two ideals $B$ and $C$ with $B+C=R$ and $B \cap C=0$, then $B$ and $C$ are rings and $R\cong B\times C$
I tried to prove it using definition of subrings, that is, B and C is closed under identity, subtraction and multiplication. But then I encountered a problem: If they are rings, then they both contain the multiplicative unit 1, but it contradicts with the fact that B intersect C is 0.
If they have different multiplicative units, then they can't be subrings of R since the unit is not preserved. I'm stuck on this....
What do you guys suggest? Any help will be appreciated!