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# 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*

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