Abstract Algebra – Generators of Finitely Generated Free Module Over Commutative Ring

Question

Let $L$ be a finitely generated free module over a commutative ring $A$. Set $n=\operatorname{rank} L$. Let $x_1,\dots,x_m$ be generators of $L$. Then $m \ge n$? If $m = n$, then is $x_1,\dots,x_m$ a basis of $L$?

Answer 1

If $x_1,\ldots,x_m$ generate $L$, then you get a surjective $A$-module map $A^m\rightarrow L$. Tensoring with $k(\mathfrak{m})=A/\mathfrak{m}$, $\mathfrak{m}$ a maximal ideal, gives you a surjection from an $m$-dimensional $k(\mathfrak{m})$-vector space to an $n$-dimensional $k(\mathfrak{m})$-vector space, so $m\geq n$.

If $n=m$, then you get a surjective endomorphism $L\rightarrow L$, and any surjective endomorphism of a finite $A$-module is injective. So in this case the elements form a basis.