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'injective module'
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| Title of object: |
injective module |
| Canonical Name: |
InjectiveModule |
| Type: |
Definition |
| Created on: |
2001-12-12 00:00:50 |
| Modified on: |
2002-01-05 15:57:42 |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}
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Content:
A module $Q$ is {\it injective}
if it satisfies the following equivalent conditions:
(a) Every short exact sequence
of the form $0 \to Q \to B \to C \to 0$
is \PMlinkname{split}{SplitShortExactSequence};
(b) The functor ${\rm Hom}(-, Q)$
is \PMlinkname{exact}{ExactFunctor};
(c) If $f : X \to Y$ is a monomorphism
and there exists a homomorphism $g : X \to Q$,
then there exists a homomorphism $h : Y \to Q$
such that $hf = g$.
\xymatrix{
\ar[r]
&
\ar[d]_g
\ar[r]^f
&
\ar@{-->}[dl]^h
\\
&
$$ |
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