Let be a ring with 1 and let be a family of ideals in indexed by a set ( need not be finite for the moment.) Let be a unital left -module. Prove that the mapping given by is an -module homomorphism with kernel .
[Note: If you’re following along with Dummit & Foote, you may notice that we jumped ahead a bit by assuming that is naturally an -module. It is, and this result could have been proved shortly after the definition of module, so we will let slide the fact that we haven’t yet formally proven that the product of modules is a module.]
First, note that for all and , we have . Thus is an -module homomorphism.
Now suppose . Then for all , we have , and thus . So . Conversely, if , then for all . Thus .