Corollary

— Let E be a module over a principal ideal domain A, generated by n elements. Then any submodule F of E can be generated by at most n elements. Indeed there exists a homomorphism f from $A^n$ onto E (II, p. 216, Cor. 3), and $f^{-1}(F)$, which is a free module of rank $m \leq n$, is generated by n elements; the images of these elements under f generate F.