---
tag: YP5TQ6WFVN
kind: concept
name: Nonassociative ring
msc: [17A01]
sources:
  - url: https://en.wikipedia.org/wiki/Non-associative_algebra
---

## Summary

A nonassociative ring is an abelian group with a multiplication that distributes over addition on both sides, without the requirement that it be associative.
"Nonassociative" means "not necessarily associative", so associative rings are a special case.
Lie algebras, [[Jordan algebras|6C2MCVB8C7]] and the [[octonions|1HRMT3DSX2]] are the main examples where associativity fails.
A nonassociative algebra over a field is the same notion with a compatible vector space structure.

## Explanations

### L1 human

In most familiar number systems, $(ab)c = a(bc)$, so brackets in products do not matter.
A nonassociative ring drops this rule and keeps the others: addition works as usual, and multiplication distributes over addition.
The cross product of vectors in space is an example: $(e_1 \times e_1) \times e_2 = 0$, but $e_1 \times (e_1 \times e_2) = e_1 \times e_3 = -e_2$.
Such products are typical in geometry and physics, where they measure rotations or symmetries rather than repeated scaling.