PlanetMath: projective module

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projective module

(Definition)

A module is projective if it satisfies the following equivalent conditions:

(a) Every short exact sequence of the form $0 \to A \to B \to P \to 0$ is split;

(b) The functor ${\rm Hom}(P, -)$ is exact;

(c) If $f : X \to Y$ is an epimorphism and there exists a homomorphism $g : P \to Y$ , then there exists a homomorphism $h : P \to X$ such that .

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & P \ar@{-->}[dl]_h \ar[d]^g \ X \ar[r]_f & Y \ar[r] & 0 } } \end{xy}$

(d) The module is a direct summand of a free module.

"projective module" is owned by antizeus.

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Cross-references: free module, direct summand, epimorphism, functor, short exact sequence, equivalent, module
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This is version 3 of projective module, born on 2002-01-05, modified 2003-09-20.
Object id is 1365, canonical name is ProjectiveModule.
Accessed 4862 times total.

Classification:

16D40 (Associative rings and algebras :: Modules, bimodules and ideals :: Free, projective, and flat modules and ideals)

Pending Errata and Addenda

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