Let $R$ be a commutative ring and let $M,N$ be free modules over $R$ and suppose we have a map $f: M \rightarrow N$ such that upon taking $\text{ Hom}(-,R)$ we get an isomorphism $f^* : N^* \rightarrow M^*.$ Must $f$ be an isomorphism as well? This is clear if $M,N$ are free and of […]

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 $L$?

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