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 that if you have a short exact sequence where one of the modules is injective, it allows for certain extensions and lifting properties that are crucial in homological algebra. The concept connects deeply with projective modules and plays a significant role in constructing projective and injective resolutions, understanding exact sequences, and utilizing the Ext functor effectively.

5 Must Know Facts For Your Next Test

1. Injective modules allow for the extension of homomorphisms from submodules to the entire module, making them vital in defining certain properties in homological algebra.
2. Every injective module is a flat module, meaning it preserves exactness when tensored with any other module.
3. Injective modules are characterized by the property that every short exact sequence that ends with an injective module splits.
4. In a Noetherian context, any injective module can be represented as a direct sum of indecomposable injective modules.
5. The category of left R-modules has enough injectives if R is a ring; this means you can find an injective module that can help resolve other modules.

Review Questions

• How does the property of an injective module facilitate the extension of homomorphisms in exact sequences?
  • Injective modules have the crucial property that any homomorphism from a submodule can be extended to the whole module. This extension property ensures that if we have a short exact sequence involving an injective module, it allows us to analyze and construct various homomorphic images and preimages. Thus, injective modules help maintain exactness in sequences and allow for decomposing complex structures into simpler components.
• Discuss the similarities and differences between injective and projective modules regarding their lifting properties.
  • Both injective and projective modules exhibit specific lifting properties essential in homological algebra. While injective modules allow for extensions of homomorphisms from submodules, projective modules allow for lifting of epimorphisms. The key difference lies in their roles: injective modules facilitate homomorphism extensions, whereas projective modules focus on lifting mappings related to surjective maps. This distinction helps us understand how each type contributes to resolutions and exact sequences.
• Evaluate how the concept of injective modules enhances our understanding of the Ext functor and its applications in classifying modules.
  • Injective modules play a significant role in understanding the Ext functor because they provide insights into how well other modules can be extended or resolved. The Ext functor helps measure how non-projective a module is by examining its relationships with injectives. Therefore, knowing about injective modules allows us to classify extensions more effectively, which leads to deeper results about module theory and impacts various branches of mathematics, including representation theory and algebraic geometry.

Related terms

Projective Module: A projective module is a module that satisfies a lifting property similar to injective modules, but in the opposite direction, allowing for certain epimorphisms to be lifted.

Exact Sequence: An exact sequence is a sequence of modules and homomorphisms where the image of one homomorphism equals the kernel of the next, reflecting important relationships between modules.

Ext Functor: The Ext functor measures the extent to which a module fails to be projective, providing information about extensions and classifying different types of modules.