Let be a ring and let be a family of left -modules indexed by a (nonempty) set . Prove that is a left -module via the action (Called the direct product of the modules ). Moreover, prove that the subset is a submodule of (Called the direct sum of the modules ). Moreover, show that and are not isomorphic as -modules.
We saw in this previous exercise that is an abelian group under pointwise addition and multiplication. Thus it suffices to show that the three left-module axioms are satisfied. To that end, let and let . We have , , and . Thus is a left -module. Suppose further that has a 1 and that each is unital; then , for all , so that is unital.
Now we will use the submodule criterion to show that is a submodule of . To that end, note first that is nonempty since . (The set of indices such that is nonzero is empty, which is certainly finite). Now let and let . Say such that for and for . Consider . Note that if , then . Since is finite, . By the submodule criterion, is an -submodule.
Now consider and as -modules in the natural way. We intend to show that these two modules are not isomorphic by showing that one is torsion while the other is not. To that end, let . There is a natural number such that, for all , . Now let (where this product is taken over the natural numbers). Certainly then ; that is, every element of is torsion, so that is torsion. However, note that is nonzero for all integers since (for instance) the component of is 1. So is not torsion. Thus and are not isomorphic as -modules.