That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative.
Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation.
For example, the octonions are unital, but Lie algebras never are.
More generally, some authors consider the concept of a non-associative algebra over a commutative ring R: An R-module equipped with an R-bilinear binary multiplication operation.
[2] If a structure obeys all of the ring axioms apart from associativity (for example, any R-algebra), then it is naturally a
Ring-like structures with two binary operations and no other restrictions are a broad class, one which is too general to study.
For this reason, the best-known kinds of non-associative algebras satisfy identities, or properties, which simplify multiplication somewhat.
The center of A is the set of elements that commute and associate with everything in A, that is the intersection of with the nucleus.
Unlike the associative case, elements with a (two-sided) multiplicative inverse might also be a zero divisor.
For example, all non-zero elements of the sedenions have a two-sided inverse, but some of them are also zero divisors.
The free non-associative algebra on a set X over a field K is defined as the algebra with basis consisting of all non-associative monomials, finite formal products of elements of X retaining parentheses.
The algebra is unital if one takes the empty product as a monomial.
An algebra is central if its centroid consists of the K-scalar multiples of the identity.
[16] Some of the possible identities satisfied by non-associative algebras may be conveniently expressed in terms of the linear maps:[36] The quadratic representation Q is defined by[37] or equivalently, The article on universal enveloping algebras describes the canonical construction of enveloping algebras, as well as the PBW-type theorems for them.