Let be an integral domain and let and be (left, unital) -modules. Prove that , where the rank of a module is the largest possible cardinality of a linearly independent subset.
Suppose has rank and has rank . By the previous exercise, there exist free submodules and having free ranks and , respectively, such that the quotients and are torsion. Note that is free. By this previous exercise, we have . Note that since is an integral domain, finite direct sums of torsion modules are torsion. Thus is torsion. Since is free and has free rank , by this previous exercise, has rank .