Fulton 2.11

2.54. What does $M$ being free on $m_1,\ldots,m_n$ say in terms of the elements of $M$ ?

It means that every element of $M$ may be written as a sum $\sum_{i=1}^na_im_i,$ where $a_1,\ldots,a_n\in R.$

2.55. Let $F=X^n+a_1X^{n-1}+\ldots+a_n$ be a monic polynomial in $R[X].$ Show that $R[X]/(F)$ is a free $R$ -module with basis $\overline{1},\overline{X}, \ldots,\overline{X}^{n-1},$ where $\overline{X}$ is the residue of $X.$

Let $X=\{\overline{1},\overline{X},\ldots,\overline{X}^{n-1}\}.$ Observe that the natural $R$ -module homomorphism $M_X\to R[X]/(F)$ is surjective with trivial kernel, and thus an isomorphism.

2.56. Show that a subset $X$ of a module $M$ generates $M$ if and only if the homomorphism $M_X\to M$ is onto. Every module is isomorphic to a quotient of a free module.

The first statement is immediate from Problem 2.54. Let $M$ be an arbitrary $R$ -module and $X$ a (possibly infinite) set of generators for $M.$ Then there exists a natural surjective homomorphism $\pi:M_X\to M$ and $M_X/\ker\pi$ is isomorphic to $M.$