Let be a ring and let be a left -module. Let be a family of submodules of indexed by a nonempty set . Prove that the following are equivalent:
- The mapping given by is an injective -module homomorphism whose image is .
- If is a (nonempty) finite subset and , then .
- If is a (nonempty) finite subset, then . (Where this denotes a finite internal direct product.)
- For every element , there exist unique elements such that all but finitely many are zero and .
We will use the following strategy: .
: Suppose is injective and that its image is the submodule generated by the . Now let be a finite nonempty subset and let . Suppose now that . Say and , where . Let if and otherwise, and let if and 0 otherwise. Certainly , and moreover . So . Comparing entries, we see that for all , and hence . Thus as desired.
Suppose is a finite nonempty subset. Note that is one of the equivalent conditions that define what it means for the internal sum to be direct; thus .
Let . By definition, there exist such that finitely many are nonzero and . Suppose now that there exist such that finitely many are nonzero and . Let be the set of all indices such that either or is nonzero. Now ; thus we have for all , so that the expansion is unique.
Define by . is clearly an -module homomorphism. Now if , then . Since the sum over the (finite number of) indices such that or is direct, we have for all . Thus , and so is injective. It is clear that the image of is .
Extending our notion of (finite) internal direct sums, if is a family of submodules satisfying any of these equivalent conditions we will say that is the internal direct sum of the .