Injective Modules

Definition

Injective modules are a class of modules that satisfy a specific property related to homomorphisms. Specifically, a module $M$ is injective if, for any module $N$ and any injective homomorphism $f: A \to B$ of $R$-modules, every homomorphism $g: A \to M$ can be extended to a homomorphism $g': B \to M$. This concept is crucial in understanding extensions and exact sequences in module theory, as it connects directly to the notions of flatness and projective modules.

5 Must Know Facts For Your Next Test

1. Injective modules play a vital role in the theory of homological algebra by allowing for the extension of homomorphisms.
2. Every divisible group is an injective abelian group, showcasing a link between injectivity and specific types of groups.
3. In the context of Noetherian rings, every injective module is also a direct sum of copies of the injective hulls of simple modules.
4. Injective modules can also be viewed through the lens of functoriality, specifically with respect to Hom functors, which illustrates their role in dualities.
5. The characterization of injective modules can be tied to the notion of Baer’s criterion, which provides conditions under which a module is injective.

Review Questions

• How do injective modules relate to homomorphisms and what significance does this have for extending mappings?
  • Injective modules are fundamentally connected to homomorphisms because they allow any homomorphism from a submodule to be extended to a larger module. This means if you have an injective module $M$, and you have a module $N$ with an injective map from $A$ to $B$, any map from $A$ to $M$ can be expanded to cover all of $B$. This property is significant in exploring module extensions and understanding how different modules interact with one another.
• Discuss how injective modules relate to flatness and projectivity within the broader framework of module theory.
  • Injective modules are related to flatness and projectivity in that they all provide insights into the structure and behavior of modules within a category. While flat modules maintain exactness under tensor products, projective modules allow for lifting homomorphisms. Injective modules provide a dual perspective where they ensure extensions are possible for homomorphisms. Together, these classes help understand how various properties can influence module interactions and facilitate easier constructions in algebraic settings.
• Evaluate the impact of Baer’s criterion on identifying injective modules and its implications for practical applications in commutative algebra.
  • Baer’s criterion gives practical conditions under which we can determine if a module is injective. Specifically, it states that a module is injective if every homomorphism from an ideal into the module can be extended to the whole ring. This criterion is particularly impactful because it provides an efficient method to identify injective modules without having to rely solely on abstract definitions. Understanding how to apply Baer's criterion facilitates advancements in commutative algebra, particularly when dealing with extensions, resolutions, and other structures that rely on injectivity.

Related terms

Flat Modules: Flat modules are those for which the tensor product with any exact sequence of modules preserves exactness, making them essential in discussions about projective resolutions.

Projective Modules: Projective modules are direct summands of free modules, and they possess the property that every homomorphism from a submodule can be lifted to a homomorphism from the entire module.

Exact Sequences: Exact sequences are sequences of modules and homomorphisms between them where the image of one homomorphism equals the kernel of the next, which helps analyze the structure of modules.