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'algebra'
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| Title of object: |
algebra |
| Canonical Name: |
Algebra |
| Type: |
Definition |
| Created on: |
2001-10-19 00:07:12-04 |
| Modified on: |
2002-05-30 12:22:12-04 |
| Classification: |
msc:16-00 |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Let $A$ be a ring. 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 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 $B$. |
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