nLab exterior ring

Idea

A $\mathbb{Z}$-exterior algebra.

Definition

Given an abelian group $G$, the exterior ring $\Lambda(G)$ is the quotient ring of the tensor ring $T(G)$ by the ideal generated by the relations $g \cdot g$ for all $g \in G$.

Universal property

Given an abelian group $G$, the exterior ring is a ring $\Lambda(G)$ with an abelian group homomorphism $g:G \to \Lambda(G)$ such that

• for every element $a:G$, $g(a) \cdot g(a) = 0$

• for every other ring $R$ with abelian group homomorphism $h:G \to R$ where for every element $a:G$, $h(a) \cdot h(a) = 0$, there is a unique ring homomorphism $i:\Lambda(G) \to R$ such that $i \circ g = h$.

See also

• ring

• tensor ring

• super ring

• Clifford ring

• supercommutative ring

• differential ring

• exterior algebra