Semisimple modules

Definition

Semisimple modules are modules over a ring that can be decomposed into a direct sum of simple modules, where a simple module is one that has no proper submodules. This property of being semisimple is closely related to the structure of Artinian rings, as every semisimple module over a ring corresponds to a specific type of decomposition that reflects the ring's ideal structure. Understanding semisimple modules helps in analyzing the behavior of modules over Artinian rings and their relationship with Noetherian rings.

5 Must Know Facts For Your Next Test

1. A ring is semisimple if it is a direct product of simple Artinian rings, which implies that every module over it is semisimple.
2. All semisimple modules are projective and injective, meaning they exhibit nice properties when dealing with extensions and resolutions.
3. In the context of Artinian rings, every submodule of a semisimple module is also semisimple, which is useful for understanding module structure.
4. The Wedderburn-Artin theorem states that every semisimple ring is isomorphic to a finite direct product of matrix rings over division rings.
5. Semisimple modules allow for a well-defined representation theory, facilitating the study of linear transformations and their properties.

Review Questions

• How do semisimple modules relate to simple modules and what implications does this have for their structure?
  • Semisimple modules are built from simple modules, which are their fundamental components. This relationship implies that any semisimple module can be expressed as a direct sum of simple modules. Consequently, this structure leads to easy decomposition and analysis of semisimple modules, making them particularly manageable in terms of understanding their behavior and properties within the broader context of ring theory.
• Discuss the significance of Artinian rings in the study of semisimple modules and how this connection aids in module classification.
  • Artinian rings play a crucial role in the study of semisimple modules because they guarantee certain structural properties that facilitate classification. Since all finite-dimensional representations over Artinian rings are semisimple, this connection allows mathematicians to classify all modules over such rings as direct sums of simple modules. Moreover, since every submodule of a semisimple module is also semisimple, this further simplifies understanding their composition and interaction within Artinian frameworks.
• Evaluate the impact of the Wedderburn-Artin theorem on the understanding of semisimple rings and their corresponding module theory.
  • The Wedderburn-Artin theorem provides a powerful insight into semisimple rings by stating that they can be decomposed into a finite direct product of matrix rings over division rings. This theorem significantly impacts module theory because it allows for a clear characterization of semisimple modules: they can be understood through their representations as matrices over division rings. This means that studying these modules not only sheds light on their algebraic structure but also connects them to linear algebra concepts, providing tools for further exploration in representation theory.

Related terms

simple modules: Simple modules are non-zero modules that have no proper submodules, serving as the building blocks for semisimple modules.

Artinian rings: Artinian rings are rings that satisfy the descending chain condition on ideals, leading to certain structural properties, including the presence of semisimple modules.

Noetherian rings: Noetherian rings are rings that satisfy the ascending chain condition on ideals, and their relationship with semisimple modules helps understand the duality between these two types of rings.