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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''.


In this article, all rings are assumed to be [[unital algebra|unital]].


==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]] [·,&nbsp;·]
 
: <math>[\cdot,\cdot]: A
\times A\to A</math>
 
called ''A''-'''multiplication''', which satisfies the following axiom:
 
* [[Bilinear operator|Bilinearity]]:
 
::<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>
 
:for all scalars ''a'', ''b'' in ''R'' and all elements ''x'', ''y'', ''z'' in ''A''.
 
==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''.
 
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.)
 
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.
 
==See also==
* [[Abelian algebra]]
* [[Algebraic structure]] (a much more general term)
* [[Associative algebra]]
* [[Coalgebra]]
* [[Graded algebra]]
* [[Lie algebra]]
* [[Semiring]]
* [[Split-biquaternion]] (example)
* [[Example of a non-associative algebra]] (example)
 
==References==
*{{Lang Algebra}}
 
[[Category:Algebras| ]]
[[Category:Ring theory]]

Revision as of 21:44, 24 February 2014


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