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algebra (Definition)

In this definition, all rings are assumed to be rings with identity and all ring homomorphisms are assumed to be unital.

Let $ A$ be a ring. An algebra over $ A$ is a ring $ B$ together with a ring homomorphism $ f\colon A \to Z(B)$, where $ Z(B)$ denotes the center of $ B$. A subalgebra of $ B$ is a subset of $ B$ which is an algebra.

Equivalently, an algebra over $ A$ is an $ A$-module $ B$ which is a ring and satisfies the property

$\displaystyle a\cdot(x*y) = (a\cdot x)*y = x*(a\cdot y)$
for all $ a \in A$ and all $ x,y \in B$. Here $ \cdot$ denotes $ A$-module multiplication and $ *$ denotes ring multiplication in $ B$. One passes between the two definitions as follows: given any ring homomorphism $ f\colon A \longrightarrow Z(B)$, the scalar multiplication rule
$\displaystyle a \cdot b := f(a)*b $
makes $ B$ into an $ A$-module in the sense of the second definition. Conversely, if $ B$ satisfies the requirements of the second definition, then the function $ f\colon A \to B$ defined by $ f(a) := a \cdot 1$ is a ring homomorphism from $ A$ into $ Z(B)$.



"algebra" is owned by djao.
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Also defines:  subalgebra

Attachments:
algebra (module) (Definition) by mathcam
every ring is an integer algebra (Example) by rspuzio
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Cross-references: function, scalar, definitions, ring multiplication, multiplication, property, subset, center, unital, ring homomorphisms, identity, rings
There are 183 references to this entry.

This is version 10 of algebra, born on 2001-10-19, modified 2006-07-31.
Object id is 353, canonical name is Algebra.
Accessed 23268 times total.

Classification:
AMS MSC: 13B02 (Commutative rings and algebras :: Ring extensions and related topics :: Extension theory)
 16S99 (Associative rings and algebras :: Rings and algebras arising under various constructions :: Miscellaneous)
 20C99 (Group theory and generalizations :: Representation theory of groups :: Miscellaneous)
Pending Errata and Addenda
None.
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algebra by remag12 on 2007-08-18 16:29:04
In your definition of an algebra, there are many authors, who do not require that the ring has an identity at all. The Bourbaki school does, but as mentioned many mathematicians do not. The presence of an identity element helps, when maximal ideals (or one sided ideals are needed of course.)
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for the people, by the people by jasonjayr on 2001-11-06 22:54:07
While this entry may be technically correct, I still woulden't know what algebra was by reading it ;) May I suggest a short history of the invention of algebra, as well something to to tune of "a system to manipulate mathematical formulas"
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