Let $R$ be a unital associative ring and
$$ R=A\oplus B, $$
where $A$ is a two-sided ideal of $R$ and $B$ is a right ideal of $R$. Does it follow that $B$ is a two-sided ideal of $R$?
I feel like the answer should be yes. Say I take an arbitrary element $(r,s)\in R$, I need to show that $(r,s)b\in B$ for all $b\in B$. But an element of $B$ can only be written as $(0,b)$ for some $b\in R$, so
$$ (r,s)(0,b)=(0,sb)\in B. $$
I feel like I am missing something and it's not that simple, since I'm not using the fact that $R$ is unital at all. What am I missing here?