Injective Module

Definition

An injective module is a type of module that satisfies the property that any homomorphism from a submodule can be extended to the entire module. This means that if you have a module $M$ and an injective module $I$, any morphism defined on a submodule of $M$ can be lifted to $I$. Injective modules play a crucial role in homological algebra, particularly in the context of Ext and Tor functors, where they help characterize projective resolutions and modules that are 'nice' with respect to exact sequences.

5 Must Know Facts For Your Next Test

1. Injective modules can be characterized using the Baer criterion, which states that a module is injective if every homomorphism from an ideal to the module can be extended to a homomorphism from the ring.
2. Every divisible abelian group is an injective abelian group, making them examples of injective modules in the category of abelian groups.
3. In the context of commutative rings, the field of fractions of an integral domain is an example of an injective module over that domain.
4. Injective modules are used to study the Ext functor, particularly in computing Ext groups, which measure the extent to which a module fails to be projective.
5. The class of injective modules is closed under taking direct sums, meaning that if you take any collection of injective modules, their direct sum will also be injective.

Review Questions

• What properties define an injective module, and how do these properties relate to morphisms and submodules?
  • An injective module is defined by its ability to extend homomorphisms from submodules to itself. This means if there's a homomorphism from a submodule to an injective module, it can always be extended to a homomorphism from the entire module. This property makes injective modules essential in situations where you need to lift mappings without losing structure, particularly when dealing with exact sequences.
• Discuss how injective modules interact with the Ext functor and why this interaction is significant in homological algebra.
  • Injective modules play a key role in the theory of the Ext functor, as they provide a way to compute Ext groups. These groups help identify whether a given module has certain projective properties. When examining exact sequences, injective modules allow for the lifting of morphisms and thus help resolve questions about extensions and cohomology in various algebraic contexts. Their ability to serve as 'targets' for extensions makes them indispensable in understanding module categories.
• Evaluate the implications of using injective modules when constructing projective resolutions and analyzing module categories.
  • Using injective modules in constructing projective resolutions allows for clearer insights into the relationships between different modules in a category. Their lifting property enables us to define extensions neatly, which is critical when investigating how various modules fit together within a category. By analyzing how injective modules interact with both projective and free modules, we can gain deeper understanding of cohomological dimensions and their associated invariants, leading to significant insights into the structure and classification of modules.

Related terms

Projective Module:

A projective module is one that satisfies the property that any homomorphism from it to another module can be lifted along surjective homomorphisms.

Exact Sequence:

An exact sequence is a sequence of modules and homomorphisms between them such that the image of one homomorphism equals the kernel of the next.

Homomorphism:

A homomorphism is a structure-preserving map between two algebraic structures, such as groups, rings, or modules.