Exercise 16, § 11

Let E be a free A-module of dimension $n$. The regressive product of $x \in \bigwedge(E)$ and $y \in \bigwedge(E)$ with respect to a basis $\{e\}$ of $\bigwedge^n(E)$, denoted by $x \vee y$, is the element $\phi(\phi(x) \wedge \phi(y))$, the isomorphism $\phi$ being relative to $e$. This product is only defined to within an invertible factor, depending on the basis $\{e\}$ chosen. Show that if $x$ is a $p$-vector and $y$ a $q$-vector, $x \vee y = 0$ if $p + q < n$ and that, if $p + q \geq n$, $x \vee y$ is a $(p + q - n)$-vector such that

The regressive product is associative and distributive with respect to addition and defines on $\bigwedge(E)$ a unital algebra structure isomorphic to the exterior algebra structure. Express the components of $x \vee y$ as a function of those of $x$ and $y$ for a given basis on E.