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. a + b = b + a
   2. (a + b) + c = a + (b + c)
   3. There exists 0 in R such that 0 + a = a + 0 = 0
   4. For each a in R, there exists an element -a such that a + ( - a) = ( - a) + a = 0
2. Multiplication is linear in each variable:
   1. (a + b)c = ac + bc (left distributive law)
   2. a(b + c) = ab + ac (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 1x = x1 = x.

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.

Examples

The octonions, constructed by John T. Graves in 1843, were the first example of a nonassociative ring that is not associative. The hyperbolic quaternions of Alexander MacFarlane form a nonassociative ring that suggested the mathematical footing for spacetime theory that followed later.

Other examples of nonassociative rings include the following:

• The Cayley-Dickson construction provides an infinite family of nonassociative rings.
• Lie algebras
• Jordan algebras.

Note that all of the above examples are actually algebras.

Properties

Most elementary properties of rings fail in the absence of associativity. For example, for a nonassociative ring with an identity element:

• If an element x has left and right inverses, a^L and a^R, then a^L and a^R can be distinct.
• Elements with multiplicative inverses can still be zero divisors.