Injective object

Injective object is a concept from the mathematical branch of category theory.

  • 4.1 Examples

Definition

In one category, an object is called injective if, for every monomorphism and every one, so that is.

Thus is injective if the induced map is surjective for all monomorphisms.

Examples

  • In the category of sets Me lot is injective.
  • Injective objects in the category of abelian groups are the divisible groups, ie those groups for which the multiplication by an integer equal to zero is surjective; Examples are and.
  • In the category of vector spaces over a field of each object is injective.
  • Each terminal object in a category is injective.
  • Is a family of injective objects, then the product of this family is injective, if it exists.
  • If the category is a zero object, a product of injective objects is injective if each is injective.
  • Is injective, then every monomorphism is a section (That is, there is a with ).
  • In the category of topological spaces, the amount is not injective, because the inclusion mapping is not a cut. There is no continuous surjective function. This is a consequence of the intermediate value theorem.

Injective modules

For a right module over a ring the following statements are equivalent.

Examples

There are a sufficient number of injective modules

Each module can be monomorphic displayed in an injective module.

Injective hull

A sub-module is large when the single sub-module is, which has the average. A monomorphism is called essential if it is large in. The following applies:

Each module can be mapped much in an injective module. The module is uniquely determined by this property, up to isomorphism. It is called injective hull of M and is often referred to.

Indecomposable injective modules

A module is called directly indecomposable if it is not a direct sum of two submodules nonzero. For a module, the following statements are equivalent.

A module that satisfies the equivalent properties of the sentence is called uniform. is then often called irreducible ( durchschnittsirreduzibel ).

Examples

  • Every simple module is uniform, ie has a directly indecomposable injective hull.
  • Is a prime ideal in the commutative ring, so is uniform integrity particular, each ring is uniform as a module.
Mathematics Reinhold Baer Semisimple module Hereditary ring Local ring
412839
de