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 $\map \bigwedge M$.

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We call:

$\map {\bigwedge^k} M$

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

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

Also see

• Definition:Alternative Algebra
• Definition:Clifford Algebra
• Definition:Geometric Algebra
• Definition:Symmetric Algebra

Sources

• 2003: Joseph J. Rotman: Advanced Modern Algebra (2nd ed.)
• 2004: David S. Dummit and Richard M. Foote: Abstract Algebra (3rd ed.)

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