article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification...

non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative....

multiplication are associative, but not (generally) commutative. Associative operations are abundant in mathematics; in fact, many algebraic structures (such...

multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras where associativity of multiplication is assumed...

where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with...

yb. For associative algebras, the definition can be simplified as follows: a non-zero associative algebra over a field is a division algebra if and only...

. Every associative algebra is power-associative, but so are all other alternative algebras (like the octonions, which are non-associative) and even...

algebra of observables of any quantum system should be a formally real algebra that is commutative (xy = yx) and power-associative (the associative law...

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have x...

c)-(a\cdot b)\cdot c=0} . But, there are algebras R {\displaystyle R} which are not necessarily associative, meaning if a , b , c ∈ R {\displaystyle a...

In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements x, y we...

{\displaystyle [x,y]} . A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same...

In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital)...

dimension 8 over F. In other words, it is a 8-dimensional unital non-associative algebra A over F with a non-degenerate quadratic form N (called the norm...

In mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N...

In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also...

may be regarded as a free commutative algebra. For R a commutative ring, the free (associative, unital) algebra on n indeterminates {X1,...,Xn} is the...

which there is no requirement for multiplication to be associative. For these authors, every algebra is a "ring". The most familiar example of a ring is...

Okubo algebras are composition algebras, flexible algebras (A(BA) = (AB)A), Lie admissible algebras, and power associative, but are not associative, not...

mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure...

algebras General Associative property, Associator Heap (mathematics) Magma (algebra) Loop (algebra), Quasigroup Nonassociative ring, Non-associative algebra...

• yy ≡ xy • y Power-associative If the submagma generated by any element is associative Flexible if xy • x ≡ x • yx Associative If it satisfies the identity...

and associative then B is associative. If A is associative and x + x*, xx* associate and commute with everything, then B is an alternative algebra. Schafer...

abstract algebra, a quasi-free algebra is an associative algebra that satisfies the lifting property similar to that of a formally smooth algebra in commutative...

enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal...

project began in 1872 when Benjamin Peirce first published his Linear Associative Algebra, and was carried forward by his son Charles Sanders Peirce. Most...

In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative...

category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams...

antipode to create a Hopf algebra structure. Note: In this article, all algebras are assumed to be unital and associative. The unit is explicitly required...

n = 1 , … , k {\displaystyle R\ /\ I_{n},\;n=1,\ldots ,k} ⁠. An associative algebra A {\displaystyle A} over a commutative ring R {\displaystyle R} is...