A non-associative algebra [1] or distributive algebra is an algebra over a field where the binary multiplication operation is not assumed to be associative. Examples include Lie algebras , Jordan algebras , the octonions , and three-dimensional Euclidean space equipped with the cross product operation.

Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions ab cd , a bc d and a b cd may all yield different answers. While this use of non-associative means that associativity is not assumed, it does not mean that associativity is disallowed.

abstract algebra - Non-associative, non-commutative binary operation with a identity - Mathematics Stack Exchange

In other words, "non-associative" means "not necessarily associative", just as "noncommutative" means "not necessarily commutative" for noncommutative rings. For example, the octonions are unital, but Lie algebras never are. The nonassociative algebra structure of A may be studied by associating it with other associative algebras which are subalgebra of the full algebra of K - endomorphisms of A as a K -vector space.

Two such are the derivation algebra and the associative enveloping algebra , the latter being in a sense "the smallest associative algebra containing A ".

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.

Associative - definition of associative by The Free Dictionary

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 which simplify multiplication somewhat.

These include the following identities. The nucleus is the set of elements that associate with all others: There are several properties that may be familiar from ring theory, or from associative algebras, which are not always true for non-associative algebras.

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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 product of monomials u , v is just u v. The algebra is unital if one takes the empty product as a monomial. Kurosh proved that every subalgebra of a free non-associative algebra is free.

An algebra A over a field K is in particular a K -vector space and so one can consider the associative algebra End K A of K -linear vector space endomorphism of A. We can associate to the algebra structure on A two subalgebras of End K A , the derivation algebra and the associative enveloping algebra. A derivation on A is a map D with the property. The derivations on A form a subspace Der K A in End K A.

The commutator of two derivations is again a derivation, so that the Lie bracket gives Der K A a structure of Lie algebra. There are linear maps L and R attached to each element a of an algebra A: The associative enveloping algebra or multiplication algebra of A is the associative algebra generated by the left and right linear maps.

An algebra is central if its centroid consists of the K -scalar multiples of the identity. Some of the possible identities satisfied by non-associative algebras may be conveniently expressed in terms of the linear maps: The quadratic representation Q is defined by [25]. The article on universal enveloping algebras describes the canonical construction of enveloping algebras, as well as the PBW-type theorems for them. For Lie algebras, such enveloping algebras have a universal property, which does not hold, in general, for non-associative algebras.

The best-known example is, perhaps the Albert algebra , an exceptional Jordan algebra that is not enveloped by the canonical construction of the enveloping algebra for Jordan algebras. From Wikipedia, the free encyclopedia.

This article is about a particular structure known as a non-associative algebra. For non-associativity in general, see Non-associativity. Ring Semiring Near-ring Commutative ring Integral domain Field Division ring. Module Group with operators Vector space. Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra. Geometry of Lie groups. Mathematics and its Applications. Graduate studies in mathematics.

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