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'algebra'
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| Title of object: |
algebra |
| Canonical Name: |
Algebra |
| Type: |
Definition |
| Created on: |
2001-10-19 00:07:12 |
| Modified on: |
2005-04-10 03:17:31 |
| Classification: |
msc:13B02, msc:16S99, msc:20C99 |
Revision comment (for changes between this and next version):
Preamble:
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Content:
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 {\em algebra} over $A$ is a ring $B$ together with a ring homomorphism $f\colon A \longrightarrow Z(B)$, where $Z(B)$ denotes the center of $B$.
Equivalently, an algebra over $A$ is an $A$--module $B$ which is a ring and satisfies the property
$$a\cdot(x*y) = (a\cdot x)*y = x*(a\cdot y)$$
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
$$
a \cdot b := f(a)*b
$$
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 $f(a) := a * 1$ is a ring homomorphism from $A$ into $Z(B)$. |
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