##  [Projectivity](/projectivity) 

  ##  [Projectivity](https://algebra.quantumdictionary.io/projectivity-0) 

  

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**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A module P over a ring R is projective if it is a direct summand of a free module; equivalently P has the lifting property that every surjection M → N and map P → N lifts to a map P → M. Projectivity is a homological condition equivalent to Ext1_R(P,−) = 0.