In mathematics, an associative algebra A is a (not necessarily unital) associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.
In some areas of mathematics, associative algebras are typically assumed to have a multiplicative unit, denoted 1. To make this extra assumption clear, these associative algebras are called unital algebras. Additionally, some authors demand that all rings be unital; in this article, the word "ring" is intended to refer to potentially non-unital rings as well.
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Group-like structures
Semigroup and Monoid Quasigroup and Loop Abelian group |
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Ring-like structures
Semiring Near-ring Ring Commutative ring Integral domain Field |
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Lattice-like structures
Semilattice Lattice Map of lattices |
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Module-like structures
Group with operators Module Vector space |
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Algebra-like structures
Algebra Associative algebra Non-associative algebra Graded algebra Bialgebra |
Read more about Associative Algebra: Formal Definition, Algebra Homomorphisms, Examples, Constructions, Associativity and The Multiplication Mapping, Coalgebras, Representations
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