Divisible module

Definition

A divisible module is a type of module over a ring such that for every element in the module and every non-zero element in the ring, there exists another element in the module that can be multiplied by that non-zero element to obtain the original element. This concept is important in understanding injective modules because divisible modules are examples of injective modules, which serve as a building block for more complex module theory.

5 Must Know Facts For Your Next Test

1. Divisible modules are characterized by the property that for any $m$ in the module and any non-zero $n$ in the ring, there exists some $m'$, such that $n \cdot m' = m$.
2. All divisible modules are injective, but not all injective modules are divisible; this distinction is crucial in module theory.
3. Every divisible abelian group is an example of a divisible module over the integers.
4. Divisible modules can be constructed from direct sums of copies of a given divisible group or by taking products of divisible groups.
5. The concept of divisible modules extends beyond abelian groups and can be applied to other algebraic structures, allowing for broader applications in homological algebra.

Review Questions

• How do divisible modules relate to injective modules in terms of their definitions and properties?
  • Divisible modules are a specific type of injective module that satisfy the condition where for any element and any non-zero scalar from the ring, there exists an element in the module that can 'divide' it. This property ensures that they can be used to extend homomorphisms, aligning with the definition of injective modules. Therefore, while all divisible modules are injective due to their inherent structure, injectivity encompasses more than just divisibility.
• Discuss how rational numbers serve as an example of a divisible module and its implications in understanding divisibility within modules.
  • Rational numbers as a $b{Z}$-module illustrate the concept of divisibility clearly. For any rational number and any integer, there is always another rational number that can be multiplied by that integer to yield the original number. This characteristic exemplifies the essence of divisible modules and highlights how certain sets can function not just as numbers but also within abstract algebraic frameworks. Understanding this connection allows for deeper insights into other types of modules and their properties.
• Evaluate the significance of divisible modules within the broader context of homological algebra and their impact on other concepts.
  • Divisible modules play a crucial role in homological algebra by providing foundational examples that help explore complex structures. Their connection to injective modules allows mathematicians to leverage their properties when dealing with exact sequences and resolutions. Additionally, by studying divisible modules, one gains insights into torsion theories and projective modules. Overall, they serve as pivotal tools that bridge various concepts within abstract algebra, influencing both theoretical developments and practical applications.

Related terms

injective module: A module is injective if it satisfies the property that any homomorphism from a submodule can be extended to the entire module.

simple module: A simple module is a module that has no submodules other than the zero submodule and itself, meaning it cannot be broken down further.

rational numbers as a module: The set of rational numbers forms a divisible group when viewed as a $b{Z}$-module, serving as a classic example of a divisible module.