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| In [[mathematics]], specifically in [[ring theory]], an '''algebra over a commutative ring''' is a generalization of the concept of an [[algebra over a field]], where the base [[field (mathematics)|field]] ''K'' is replaced by a [[commutative ring]] ''R''.
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| In this article, all rings are assumed to be [[unital algebra|unital]].
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| ==Formal definition==
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| Let ''R'' be a commutative ring. An ''R''-algebra is an [[module (mathematics)|''R''-module]] ''A'' together with a [[binary operation]] [·, ·]
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| : <math>[\cdot,\cdot]: A
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| \times A\to A</math>
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| called ''A''-'''multiplication''', which satisfies the following axiom:
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| * [[Bilinear operator|Bilinearity]]:
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| ::<math> [a x + b y, z] = a [x, z] + b [y, z], \quad [z, a x + b y] = a[z, x] + b [z, y] </math>
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| :for all scalars ''a'', ''b'' in ''R'' and all elements ''x'', ''y'', ''z'' in ''A''.
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| ==Associative algebras== | |
| If ''A'' is a [[monoid]] under ''A''-multiplication (it satisfies associativity and it has an identity), then the ''R''-algebra is called an [[associative algebra]]. An associative algebra forms a ring over ''R'' and provides a generalization of a ring. An equivalent definition of an associative ''R''-algebra is a ring homomorphism <math>f:R\to A</math> such that the image of ''f'' is contained in the center of ''A''.
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| Alternative definition: Given a ring homomorphism <math>\lambda: A \to B</math> we say that ''B'' is an ''A''-algebra. (Matsumura, Commutative Ring Theory, p 269.)
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| A ring homomorphism <math>\rho: A \to B</math> shall always map the identity of ''A'' to the identity of ''B''. We also say that ''B''/''A'' is an algebra over A given by <math>\rho</math>. Every ring is a <math>\mathbb{Z}</math>-algebra. Kunz, Intro, Conventions.
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| ==See also==
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| * [[Abelian algebra]]
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| * [[Algebraic structure]] (a much more general term)
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| * [[Associative algebra]]
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| * [[Coalgebra]]
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| * [[Graded algebra]]
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| * [[Lie algebra]]
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| * [[Semiring]]
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| * [[Split-biquaternion]] (example)
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| * [[Example of a non-associative algebra]] (example)
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| ==References==
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| *{{Lang Algebra}}
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| [[Category:Algebras| ]]
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| [[Category:Ring theory]]
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