nLab exterior ring

Contents

Context

Algebra

Group theory

Contents

Idea

A \mathbb{Z}-exterior algebra.

Definition

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

Universal property

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

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

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

See also

Last revised on May 20, 2022 at 20:10:51. See the history of this page for a list of all contributions to it.