finitely generated & finitely related = finitely presented module? Let $R$ be a ring $M$ an $R$-module. How can I prove that if 


*

*$M\cong R^n/N$ for some $n\!\in\!\mathbb{N}$ and some submodule $N\leq R^n$ 


and if 


*

*$M\cong R^{(I)}/\langle u_1,\ldots,u_m\rangle$ for some set $I$ and some vectors $u_1,\ldots,u_m\in R^{(I)}$,


then


*

*$M\cong R^k/\langle v_1,\ldots,v_l\rangle$ for some $k\!\in\!\mathbb{N}$ and some vectors $v_1,\ldots,v_l\in R^k$ ?


Every generating system $E$ of a finitely generated module $M$ contains a finite generating system (namely, look at those generators of $E$ which are needed to generate a finite generating system of $M$). Now assume that there are only finitely many relations between the generators of $E$. But these only use finitely many generators of $E$. It follows that every presentation of $M$ can be adjusted to a finite presentation.
