Given two-sided ideals $B$ and $C$ of a ring $A$,
(a) show that $BC \subseteq B \cap C$.
(b) If the ring $A$ is commutative and $B + C = A$, show that $BC = B \cap C$.
Here's what i have but I am unsure if it is correct.
(a) Since $A,B$ are both two-sided ideals, $aba' \in B$ and $aca' \in C$.
So, $BC = aba'(aca') = ab(a'a)ca' = abca' \subseteq B \cap C$.
(b) $B+C = aba'+aca'=a(b+c)a'$ and I got stuck.
Any help will be much appreciated.