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 .

## Comments

Perhaps I’m misreading this, but should you be taking a union instead of an intersection of the annihilators?

You’re right. This isn’t the first time I’ve mixed up ‘cap’ and ‘cup’…

Thanks!

What do you mean by fold product?

A -fold product is just a product with factors. For example, if is an ideal, then the ideal product consists of all possible finite sums of 2-fold products of elements from .

Typo in the problem: \cap should be \cup

Thanks!

Looking back, it seems I’ve fixed that typo before.