Let be an integral domain with quotient field and let be any (left, unital) -module. Prove that the rank of equals the dimension of over .
Recall that the rank of a module over a domain is the maximal number of -linearly independent elements.
Suppose is an -linearly independent set, and consider consisting of the tensors for each . Suppose . For some . Clearing denominators, we have for some . Now , and we have . By this previous exercise, there exists a nonzero such that . Since is linearly independent, we have for each , and since and is a domain, for each . Thus (since the denominators of each are nonzero). So is -linearly independent in . In particular, we have .
Now note that if , is a linearly independent set, with , then ‘clearing denominators’ by multiplying by , we have a second linearly independent set with the same cardinality whose elements are of the form for some . Suppose there exist such that ; then , and (since the form a basis) we have . So the are -linearly independent in . Thus .