projective module

In algebra, a module that is the direct summand of a free module

EntityQ942423

Subclass of: flat module
, projective object

Overview

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In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules over a ring, keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.

Overview

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29 statements · 17% with external sources · numbers in brackets are citations

Defining formula: P\in\operatorname{ProjMod}_R\iff\exists Q\in\operatorname{Mod}_R\exists\kappa\in\operatorname{Card}\colon P\oplus Q\cong R^{\oplus\kappa}
Studied by: module theory
In defining formula: R symbol represents: unital ring
P symbol represents: projective module
Q symbol represents: module
\kappa symbol represents: cardinal number
R^{\oplus\kappa} symbol represents: free module
\oplus symbol represents: direct sum of modules
\iff symbol represents: if and only if
\cong symbol represents: isomorphism
Dual notion (category theory): injective module

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Source last edited September 6, 2026. AIDb refreshes this page daily.

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AIDb. (2026). projective module (Q942423). Retrieved from https://aidb.si/e/projective-module-Q942423