Module

Definition

A module is a mathematical structure that generalizes the concept of vector spaces by allowing scalars to come from a ring instead of a field. In this way, modules can be viewed as a collection of objects that can be added together and multiplied by elements from a ring, forming a rich algebraic framework. Modules are connected to various concepts, including ring homomorphisms which preserve the module structure and tensor products that help to combine and manipulate modules in more complex ways.

5 Must Know Facts For Your Next Test

1. Modules can be finitely generated, meaning they can be expressed as linear combinations of a finite set of generators.
2. Not all modules are free; some may have relations among their generators, leading to torsion elements.
3. The category of modules over a ring is an important area in homological algebra, leading to concepts like projective and injective modules.
4. Every vector space is a module over a field, illustrating that modules generalize the concept of vector spaces.
5. Tensor products of modules allow for new constructions that can reveal deeper properties about the original modules involved.

Review Questions

• How do ring homomorphisms relate to the structure of modules?
  • Ring homomorphisms play a crucial role in understanding modules because they preserve the operations between rings and their respective modules. When you have a ring homomorphism from ring R to ring S, it induces a corresponding map between R-modules. This means that if you have an R-module, you can define an S-module using the homomorphic image, ensuring that the addition and scalar multiplication remain consistent with their respective structures.
• Discuss how submodules relate to the structure of a given module and provide an example.
  • Submodules are essential components of modules as they allow for the analysis of smaller, contained structures within a given module. A submodule is formed when you take a subset of a module that is closed under addition and scalar multiplication by elements from the ring. For example, if M is an R-module consisting of all integers under usual addition and multiplication by integers, then the set of all even integers forms a submodule of M since it satisfies both conditions.
• Evaluate how tensor products enhance our understanding of module relationships and provide an example application.
  • Tensor products significantly enrich our understanding of module relationships by providing a way to combine two modules into another module that encapsulates interactions between them. For instance, if you have two R-modules A and B, their tensor product A ⊗_R B allows you to express bilinear mappings in a structured way. An application of this concept is in homological algebra where tensor products are used to define derived functors, leading to insights in projective and injective resolutions which are foundational for studying other algebraic structures.

Related terms

Ring: A set equipped with two binary operations, usually called addition and multiplication, satisfying certain properties such as associativity, distributivity, and the existence of an additive identity.

Submodule: A subset of a module that is itself a module under the same operations as the larger module, maintaining the structure and properties of the original module.

Homomorphism: A structure-preserving map between two algebraic structures, such as modules or rings, which respects the operations defined in each structure.