Definition:Exterior Algebra

Definition
Let $M$ be an $R$-module, where $R$ is a commutative ring.

Let $\otimes$ denote the tensor product over $M$.

Then the exterior algebra of $M$ is defined as the quotient algebra $\map T M$ by the two-sided ideal $J$ such that:


 * $\map \bigwedge M = \map T M / J$

Where $J$ is a two-sided ideal generated by all elements $m \otimes m$ for all $m \in M$.

The multiplication:
 * $m_1 \wedge {m_2} \wedge \cdots \wedge {m_k}$

is known as the exterior (or wedge) product, which denotes the image of $m_1 \otimes m_2 \otimes \cdots \otimes m_k$ in $\map \bigwedge M$.

We call:
 * $\map {\bigwedge^k} M$

the $k$th exterior power of $M$, where $k \in \N$, where $\N$ denotes the natural numbers.

We may refer to $\map \bigwedge M$ as "wedge $M$".

Also see

 * Definition:Alternative Algebra
 * Definition:Clifford Algebra
 * Definition:Geometric Algebra
 * Definition:Symmetric Algebra