The module is injective iff it is a direct summand of an injective cogenerator

In summary, a module is injective if any homomorphism from a submodule can be extended to a homomorphism from the entire module. A direct summand of an injective cogenerator is a submodule that serves as a building block for the larger injective module. This statement is a characterization of injective modules and cannot be applied to other types of modules. It is useful in proving theorems and understanding the properties and structure of injective modules, which have applications in various areas of mathematics.
  • #1
ggugl
2
0
could anyone give me a proof of this statement:

The module is injective iff it is a direct summand of an injective cogenerator.
 
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  • #2
no, but for such things, see abelian categories by peter freyd.
 

What does it mean for a module to be injective?

A module is considered injective if it satisfies the property that any homomorphism from a submodule can be extended to a homomorphism from the entire module. In other words, any submodule of an injective module can be "filled in" to create a homomorphism from the entire module.

What is a direct summand of an injective cogenerator?

A direct summand of an injective cogenerator is a submodule that, when combined with other submodules, makes up the entire injective module. This submodule is also injective and serves as a building block for the larger injective module.

How does this statement relate to the concept of injective modules?

This statement is a characterization of injective modules, as it states that a module is injective if and only if it can be built from direct summands of an injective cogenerator. This provides a useful tool for identifying and understanding injective modules.

Can this statement be applied to other types of modules besides injective ones?

No, this statement specifically applies to injective modules and cannot be generalized to other types of modules.

How is this statement useful in the field of mathematics?

This statement is useful for proving theorems and making connections between different concepts in mathematics. It also helps in understanding the properties and structure of injective modules, which have applications in various areas of mathematics such as algebra and topology.

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