In some thesis there are given ideals $I_i \subset R$ which are pair comaximal and generated by central elements of ring $R$ and it's written "then $\bigcap_{i=1}^n I_i =\prod_{i=1}^n I_i$ for any $n\geq 1.$" $R$ is, of course, any noncommutative ring. This fact is used to proof another theorem stating semiprimitivity of some algebra, which I have to understand.
I know commutative version of Chinese Remainder Theorem, which requires only pair comaximality. I suppose that hypothese about generating by central elements is very important, but I have problems to obtain the proof of this fact.