Injective Module

Definition

An injective module is a type of module that has the property that any homomorphism from a submodule can be extended to the entire module. This means if you have a submodule and a map from it, you can find a corresponding map from the entire injective module. This property makes injective modules very useful in representation theory, especially when working with induction and restriction functors.

5 Must Know Facts For Your Next Test

1. Injective modules allow for the lifting of morphisms, making them essential in constructing projective resolutions.
2. Every divisible abelian group is an injective abelian group, showcasing a concrete example of an injective module.
3. Injective modules are related to the concept of exact sequences in homological algebra, where they help in analyzing extensions and equivalences between different modules.
4. In the category of modules over a ring, injective modules can be characterized using the Baer criterion, which provides a practical way to identify them.
5. The category of injective modules is closed under taking direct products and isomorphisms, which contributes to their utility in various mathematical contexts.

Review Questions

• How does the property of injectivity in modules relate to extending homomorphisms from submodules?
  • The property of injectivity in modules allows for any homomorphism defined on a submodule to be extended to a homomorphism on the entire injective module. This means if you have a map from a smaller part of the module, you can always find a way to apply that map across the whole module. This characteristic is particularly useful in representation theory when working with induction and restriction functors, allowing for seamless transitions between different representations.
• Discuss the significance of injective modules in the context of exact sequences and homological algebra.
  • Injective modules play a crucial role in homological algebra, particularly when analyzing exact sequences. In this context, they serve as an important tool for lifting morphisms and studying extensions of modules. Their ability to extend homomorphisms helps maintain the properties required for exactness in sequences, allowing mathematicians to investigate the relationships between various modules and understand their structure more deeply.
• Evaluate how injective modules contribute to the overall framework of representation theory and their implications for understanding group actions.
  • Injective modules contribute significantly to representation theory by providing a way to analyze how groups act on vector spaces or modules. Their extension property enables mathematicians to construct representations that can be lifted or modified while retaining necessary structure. By understanding these injective structures, one can gain insights into how different representations relate and interact under group actions, leading to deeper insights into both algebraic structures and their applications in various mathematical fields.

Related terms

Module: A mathematical structure where a set of elements can be added together and multiplied by scalars, similar to vector spaces but defined over rings instead of fields.

Homomorphism:

A structure-preserving map between two algebraic structures, such as groups, rings, or modules, that maintains the operations defined on those structures.

Projective Module:

A type of module that satisfies a property similar to injective modules, but it focuses on lifting homomorphisms rather than extending them, often characterized by its relation to direct summands.