Injective Module

Definition

An injective module is a type of module that has the property of being 'injective' with respect to homomorphisms. Specifically, a module $M$ is injective if, whenever there is an injective homomorphism from a submodule $N$ of some module $A$ into $M$, there exists a homomorphism from $A$ into $M$ that extends the one from $N$. This concept plays a crucial role in the study of modules and their relationships, particularly in the context of exact sequences and diagram chasing, where injective modules help to identify when certain conditions hold or can be extended.

5 Must Know Facts For Your Next Test

1. Injective modules allow for the extension of homomorphisms, meaning any map defined on a submodule can be uniquely extended to the entire module.
2. Every module can be embedded into an injective module, which makes injective modules particularly important in category theory.
3. The class of injective modules can be characterized using Baer's criterion, which provides conditions under which a module is injective based on its properties related to homomorphisms.
4. Injective modules are particularly useful in constructing resolutions in homological algebra, aiding in the computation of derived functors.
5. Over a Noetherian ring, every injective module is also a direct summand of a free module.

Review Questions

• How do injective modules relate to the concept of homomorphisms in module theory?
  • Injective modules are defined by their ability to extend homomorphisms from submodules to larger modules. This means that if you have an injective map from a submodule to an injective module, you can find a way to extend this map from the entire module. This property makes injective modules key players when working with homomorphisms and helps simplify many arguments in module theory.
• What role do injective modules play in exact sequences and how does this impact diagram chasing?
  • Injective modules are essential in exact sequences because they allow for the completion of diagrams when applying certain properties. When an exact sequence involves injective modules, one can utilize their extension property to ensure that mappings remain valid across different components of the diagram. This plays a significant role in diagram chasing, allowing mathematicians to derive important conclusions about the relationships between various modules.
• Evaluate the significance of Baer's criterion in determining whether a module is injective and its implications for other types of modules.
  • Baer's criterion provides necessary and sufficient conditions for determining whether a module is injective based on its behavior with respect to homomorphisms from ideals. The criterion states that a module is injective if every homomorphism from an ideal of the ring into the module can be extended to the entire ring. This has implications not only for understanding injective modules but also for classifying other types of modules, like projective modules, which rely on dual properties regarding surjective mappings. By understanding these criteria, one can make informed decisions about the structure and properties of various modules.

Related terms

Projective Module:

A projective module is a type of module that has a lifting property similar to injective modules but focuses on surjective homomorphisms instead.

Exact Sequence:

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

Homomorphism:

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