Nonassociative Ring

In abstract algebra, a nonassociative ring is a generalization of the concept of ring.

A nonassociative ring is a set R with two operations, addition and multiplication, such that:

1. R is an abelian group under addition:
   1. There exists 0 in R such that
   2. For each a in R, there exists an element −a such that
2. Multiplication is linear in each variable:
   1. (left distributive law)
   2. (right distributive law)

Unlike for rings, we do not require multiplication to satisfy associativity. We also do not require the presence of a unit, an element 1 such that .

In this context, nonassociative means that multiplication is not required to be associative, but associative multiplication is permitted. Thus rings, which we'll call associative rings for clarity, are a special case of nonassociative rings.

Some classes of nonassociative rings replace associative laws with different constraints on the order of application of multiplication. For example Lie rings and Lie algebras replace the associative law with the Jacobi identity, while Jordan rings and Jordan algebras replace the associative law with the Jordan identity.