## You are here

Homeenough projectives

## Primary tabs

# enough projectives

Let $\mathcal{A}$ be an abelian category. $\mathcal{A}$ is said to have *enough projectives* if, for every object $A$ of $\mathcal{A}$, there is a projective object $P$ of $\mathcal{A}$ and an exact sequence

$\xymatrix{P\ar[r]^{p}&A\ar[r]&0}.$ |

In other words, the map $p\colon P\to A$ is epi, or an epimorphism.

Example. Let $R$ be a ring. The category of left (right) $R$-modules is an abelian category having enough projectives. This is true since, for every left (right) $R$-module $M$, we can take $F$ to be the free (and hence projective) $R$-module generated by a generating set $X$ for $M$ (we can in fact take $X$ to be $M$). Then the canonical projection $\pi\colon F\to M$ is the required surjection.

More generally, a category $\mathcal{C}$ is said to have *enough projectives* if every object is a strong quotient object of a projective object.

# References

- 1
F. Borceux
*Basic Category Theory, Handbook of Categorical Algebra I*, Cambridge University Press, Cambridge (1994)

## Mathematics Subject Classification

18G05*no label found*18E10

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag

Aug 24

new question: Scheduling Algorithm by ncovella

new question: Scheduling Algorithm by ncovella