Rotman makes the following axiomatization
Definition: If $V$ is a free $k$-module of rank $n$, then a Grassmann algebra on $V$ is a $k$ algebra $G(V)$ with identity element, $e_0$, such that
- $G(V)$ contains $\langle e_0 \rangle \oplus V$ as a submodule, where $\langle e_0 \rangle \cong k$.
- $G(V)$ is generated as a $k$-algebra, by $\langle e_0 \rangle \oplus V$.
- $v^2=0$ for all $v \in V$.
- $G(V)$ is a free $k$-module of rank $2^n$.
We have Theorem 9.139, pg 747. The statement states that "the Grassmann algebra" is graded.
But the proof requires the model of a Grassman algebra constructed from part 1.
In the rest of the chapter, he also only refers to the Grassmann algebra. So question is:
Under the given axioms, are Grassmann algebras unique upto isomorphism?
In particular, why can Rotman use "the" Grassmann algebra? Or is he only using this model?