Exercise 10, § 7

Let E be an A-module with a generating system of n elements. Show that every alternating n-linear mapping of $E^n$ into an A-module F is the skewsymmetrization of an n-linear mapping of $E^n$ into F. (Let $a_j$ ($1 \leq j \leq n$) be the generators of E and $\phi : A^n \to E$ the homomorphism such that $\phi(e_j) = a_j$ for all j, where $(e_j)$ is the canonical basis of $A^n$. If $f : E^n \to F$ is an alternating n-linear mapping, so is $f_0 : (x_1, \ldots, x_n) \mapsto f(\phi(x_1), \ldots, \phi(x_n))$; show that if $g_0$ is an n-linear mapping of $(A^n)^n$ into F such that $f_0 = ag_0$, $g_0$ can also be written as $(x_1, \ldots, x_n) \mapsto g(\phi(x_1), \ldots, \phi(x_n))$, where g is an n-linear mapping of $E^n$ into F.)