An object P in a category C is projective if the hom functor
preserves epimorphisms. That is, every morphism f:P→X factors through every epi Y→X.
Let be an abelian category. In this context, an object is called a projective object if
Let be a ring with 1. Consider the category of left -modules is an abelian category. The projective objects in are precisely the projective left R-modules. So is itself a projective object in Dually, the injective objects in are exactly the injective left R-modules.
The category of left (right) -modules also has enough projectives. This is true since, for every left (right) -module , we can take to be the free (and hence projective) -module generated by a generating set for (we can in fact take to be ). Then the canonical projection is the required surjection.
- Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics 17, Academic Press, ISBN 978-0-124-99250-4, MR 0202787
This article incorporates material from Projective object on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. This article incorporates material from Enough projectives on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.