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 module and a submodule, you can always find a way to map the submodule into the injective module without losing structure. Injective modules are crucial in understanding resolutions and decompositions of modules, as they play a key role in constructing split exact sequences and understanding projective modules.

5 Must Know Facts For Your Next Test

1. Injective modules can be used to extend homomorphisms from smaller submodules, making them essential for constructing resolutions in module theory.
2. If a module is injective, then every short exact sequence that ends in it splits, which means it can be viewed as a direct summand.
3. In the category of modules over a ring, injective modules correspond to certain types of functors known as left-exact functors.
4. Every divisible abelian group is an example of an injective module, showcasing their application in various algebraic structures.
5. The use of injective modules helps in simplifying complex problems in algebra by allowing for easier handling of morphisms and extensions.

Review Questions

• How does the property of an injective module aid in extending homomorphisms, and why is this important?
  • An injective module allows for the extension of homomorphisms from submodules to the entire module, meaning that if you have a map defined on a smaller piece, you can find a way to define it on the larger piece without losing any structure. This is important because it helps maintain relationships between different modules and enables more straightforward manipulation of their properties when working with exact sequences.
• Discuss how injective modules relate to exact sequences and projective modules in terms of their structural properties.
  • Injective modules are closely tied to exact sequences because they allow such sequences to split, meaning that any sequence ending in an injective module can be decomposed into simpler components. This property contrasts with projective modules, which deal with lifting homomorphisms. Both types play vital roles in module theory but serve different purposes: injectives handle extensions while projectives manage surjective mappings.
• Evaluate the significance of injective modules in modern algebra and their applications across different mathematical contexts.
  • Injective modules are significant in modern algebra because they facilitate the construction of projective resolutions and help solve problems related to morphisms between modules. Their role extends beyond mere theoretical constructs; they apply to various areas such as homological algebra, representation theory, and even number theory. Understanding how injectives function allows mathematicians to build more complex structures while simplifying the relationships within them.

Related terms

Projective Module: A projective module is one that satisfies the property of lifting homomorphisms, meaning it can be seen as a direct summand of a free module.

Exact Sequence: An exact sequence is a sequence of modules and module homomorphisms where the image of one homomorphism equals the kernel of the next, representing how modules relate to each other.

Module Homomorphism: A module homomorphism is a function between two modules that preserves the structure of the modules, meaning it respects addition and scalar multiplication.