# nLab exterior ring

Contents

### Context

#### Algebra

higher algebra

universal algebra

group theory

# Contents

## 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 $i \circ g = h$.