Let be a ring with 1, let be a left -module, and let be an ideal of . Prove that is a submodule of .
We claim that for each , . To see this, let and let . Now by definition, is a finite sum of -fold products of elements in . Note that every -fold product of elements in is annihilated my , so that in fact . Thus .
So is the union of a chain of submodules. By this previous exercise, it is a submodule of .