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7
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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: |
2005-02-14 21:41:40 |
| Classification: |
msc:13B02, msc:16S99, msc:20C99 |
Revision comment (for changes between this and next version):
| Changes for correction #6227 ('converse'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Let $A$ be a ring\footnote{In this entry, all rings are assumed to be rings with identity and all ring homomorphisms are assumed to be unital.}. An {\em algebra} over $A$ is a ring $B$ together with a ring homomorphism $f: A \longrightarrow Z(B)$, 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 $f: A \longrightarrow Z(B)$, the scalar multiplication rule
$$
a \cdot b := f(a)*b
$$
makes $B$ into an $A$--module in the sense of the second definition. |
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