A nonassociative ring is a set R with two operations, addition and multiplication, such that:
- R is an abelian group under addition:
- There exists 0 in R such that
- For each a in R, there exists an element −a such that
- Multiplication is linear in each variable:
- (left distributive law)
- (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.
The octonions, constructed by John T. Graves in 1843, were the first example of a ring that is not associative. The hyperbolic quaternions of Alexander Macfarlane (1891) form a nonassociative ring that suggested the mathematical footing for spacetime theory that followed later.
Other examples of nonassociative rings include the following:
- (R3, +, ×) where × is the cross product of vectors in 3-space
- The Cayley–Dickson construction provides an infinite family of nonassociative rings.
- Lie algebras and Lie rings
- Jordan algebras and Jordan rings
- Alternative rings: A nonassociative ring R is said to be an alternative ring if [x,x,y] = [y,x,x] = 0, where [x,y,z] = (xy)z − x(yz) is the associator.
- Semifields (see quasifield axioms)
Most elementary properties of rings fail in the absence of associativity. For example, for a nonassociative ring with an identity element:
- If an element has left and right multiplicative inverses, and , then and can be distinct.
- Elements with multiplicative inverses can still be zero divisors.
- Susumu Okubo (1995) Introduction to Octonian and Other Non-Associative Algebras in Physics, Cambridge University Press.