Every finitely generated module is the quotient of a finitely generated projective module.

Question

Every finitely generated module is the quotient of a finitely generated projective module.

I already find a proof of this but it uses tensor functor and flat modules propositions so im looking for a simpler proof

Answer 1

In fact, it is a quotient of a finitely generated free module. Suppose that $R$ is your ring, and that $M$ is your module. Choose a generating set $\{m_1,...,m_n\}$. Define a module map $\varphi:R^n\rightarrow M$ by $$(r_1,...,r_n)\mapsto r_1m_1+\ldots+r_nm_n.$$ This map is surjective. By the First Isomorphism Theorem, $M$ is isomorphic to the quotient of $R^n$ by the kernel.