Noun
(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
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. Source: Internet
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. Source: Internet
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. Source: Internet
Even though H contains copies of the complex numbers, it is not an associative algebra over the complex numbers. Source: Internet
Indeed, they are (trivially) zero divisors and clearly form an ideal of the associative algebra (and thus ring ) of the dual numbers. Source: Internet
It can be used, however, to define the tensor product of two representations of a Lie algebra (rather than of an associative algebra). Source: Internet