Examples
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 = = 0, where = (xy)z − x(yz) is the associator.
- Semifields (see quasifield axioms)
Read more about this topic: Nonassociative Ring
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