Suppose $M$ is an $R$ module where $R$ is a ring. In Dummit and Foote Chapter 12, they claim that $M$ is finitely generated over $R$ if and only if there exists a surjective $R$-homomorphism from $R^n$ to $M$. The only if part is trivial, but in the book, we do not assume that rings contain $1$, so I can only prove the converse if I further assume that $R^n$ is finitely generated.
If I take $R$ to be the ring of polynomials vanishing at 0, then clearly $R^n$ is not finitely generated, so I fail to see how to prove the claim in the more general case. I have not found any mistakes in the book, which leads me to believe that I am either wrong or missing something subtle.