associative algebra
Noun
Meaning
(algebra) A vector space over some field which also has an associative vector-valued multiplication operator between vectors which together with the abelian group feature of the vector space makes a ring.
Synonym: linear algebra
Antonym: non-associative algebra
Hyponyms: algebraic number field, Boolean ring, tensor algebra, matrix algebra, Lie algebra, polynomial ring, polynomial algebra, Grassmann algebra, Clifford algebra, geometric algebra, central simple algebra, algebraic extension
Hypernyms: algebra over a field, non-associative algebra
(algebra) A module over a ring together with an associative bilinear module-element-valued operator between module elements which together with the abelian group feature of the module makes a ring.
Synonym: linear algebra
Antonym: non-associative algebra
Hyponyms: algebraic number field, Boolean ring, tensor algebra, matrix algebra, polynomial ring, polynomial algebra, Grassmann algebra, geometric algebra
Hypernyms: algebra over a ring, non-associative algebra
Synonyms
Examples
By definition, a ring is a monoid object in the category of abelian groups ; thus, an associative algebra is obtained by replacing the category of abelian groups with the category of modules.
For example: *An associative algebra is a ring that is also a vector space over a field K such that the scalar multiplication distributes over the ring multiplication.
For instance, in any group second powers behave well, : If the derived subgroup is central, then : Ring theory The commutator of two elements a and b of a ring or an associative algebra is defined by : It is zero if and only if a and b commute.
Even though H contains copies of the complex numbers, it is not an associative algebra over the complex numbers.
Indeed, they are (trivially) zero divisors and clearly form an ideal of the associative algebra (and thus ring ) of the dual numbers.
It can be used, however, to define the tensor product of two representations of a Lie algebra (rather than of an associative algebra).