Editing Associative algebra (section)

{{Short description|Ring that is also a vector space or a module}}
{{About|an algebraic structure|other uses of the term "algebra"|Algebra (disambiguation)}}
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In [[mathematics]], an '''associative algebra''' ''A'' over a [[commutative ring]] (often a [[Field (mathematics)|field]]) ''K'' is a [[ring (mathematics)|ring]] ''A'' together with a [[ring homomorphism]] from ''K'' into the [[center (ring theory)|center]] of ''A''. This is thus an [[algebraic structure]] with an addition,  a multiplication, and a [[scalar multiplication]] (the multiplication by the image of the ring homomorphism of an element of ''K''). The addition and multiplication operations together give ''A'' the structure of a [[ring (mathematics)|ring]]; the addition and scalar multiplication operations together give ''A'' the structure of a [[module (mathematics)|module]] or [[vector space]] over ''K''. In this article we will also use the term [[algebra over a field|''K''-algebra]] to mean an associative algebra over  ''K''. A standard first example of a ''K''-algebra is a ring of [[Square matrix|square matrices]] over a commutative ring ''K'', with the usual [[matrix multiplication]].

A '''commutative algebra''' is an associative algebra for which the multiplication is [[commutative]], or, equivalently, an associative algebra that is also a [[commutative ring]].

In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called '''unital associative algebras''' for clarification. In some areas of mathematics this assumption is not made, and we will call such structures [[unital algebra|non-unital]] associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital.

Every ring is an associative algebra over its center and over the integers.

{{Algebraic structures |Algebra}}