Let be a commutative ring with 1 and let be a finite family of ideals of . Suppose further that if . (I.e. the ideals are pairwise comaximal.) Let be a unital left -module. Prove that the -module homomorphism discussed here is surjective and has kernel , where denotes the ideal product in rather than the direct product of sets. Deduce that .
We will prove this result by induction on . For the base case , we have . First we show that is surjective. Let be in . Since and are comaximal and has a 1, there exist and such that . Note then that . Thus is surjective. Next we show that . We showed in Part 1 that . Clearly . Now suppose . Recall that for some and ; now . Since , . Since , . Thus as desired. By the First Isomorphism Theorem for modules, the induced mapping given by is an -module isomorphism.
Now for the inductive step, suppose the result holds for and . Let be a family of pairwise comaximal ideals in and let be a unital left -module. We claim that and are comaximal; to see this, note that for some and for all . Then , so that . Now .