Remark

(1) Suppose that E is a graded A-algebra of type $\mathbf{Z}$, of graduation $(E_n)$, and suppose also that the linear mapping $f$ (assumed to satisfy (1)) is such that

Then the relation $g(x_1 x_2 \ldots x_p) = f(x_1) f(x_2) \ldots f(x_p)$ with the $x_i \in M$ shows that $g(\bigwedge^p(M)) \subset E_p$ for all $p \geq 0$ and hence $g$ is a graded algebra homomorphism.

(2) To avoid confusion, the product of two elements $u, v$ of the exterior algebra $\bigwedge(M)$ is usually denoted by $u \wedge v$ and is called the exterior product of $u$ by $v$. The elements of $\bigwedge^n(M)$ are therefore the sums of elements of the form $x_1 \wedge x_2 \wedge \cdots \wedge x_n$ with $x_i \in M$ for $1 \leq i \leq n$ and are often called n-vectors.