Let be a ring. Prove that is a finitely generated module over if and only if is a module-homomorphic image of .
Suppose first that is a finitely generated -module – say by . Let denote the th standard basis vector in , and define by , and extend linearly. Certainly then is a surjective module homomorphism.
Conversely, suppose is a surjective module homomorphism, and for each let . Since is surjective, and since the generate , the set generates over as desired.