Exercise 18, § 2

Let A be a local ring, E a free A-module, and $x$ a nonzero element of E. Let $(e_i)_{i\in I}$ be a basis of E in which the number of nonzero coordinates of $x$ is the least possible. Let $x=\sum a_ie_i$ be the representation of $x$ in this basis; denote the set of indices $i\in I$ such that $a_i\not= 0$ by J.

a) Prove that none of the $a_i$ belong to the right ideal of A generated by the others. b) Let F and G be supplementary submodules of E such that $x\in F$; for $i\in I$, write $e_i=f_i+g_i$ with $f_i\in F$ and $g_i\in G$. Prove that the family consisting of the $f_j$ for $j\in J$ and the $e_i$ for $i\in I$ - J is a basis of E (if $g_j=\sum_{k\in I}b_{jk}e_k$, observe that the coordinates $b_{jk}$ for $j, k$ in J belong to the maximal ideal of A, and apply Exercise 17 d)). Deduce that there exists a free submodule of F that contains $x$ and is a direct factor of F.

c) Prove that every projective A-module is free (use Kaplansky’s theorem (II, §2, p. 386, Exercise 2) to reduce to the case of a module with a countable family of generators, and apply b)).