Semisimple module

Summary

In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings. For a group-theory analog of the same notion, see Semisimple representation. Definition A module over a (not necessarily commutative) ring is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules. For a module M, the following are equivalent:

M is semisimple; i.e., a direct sum

Official source

https://en.wikipedia.org/wiki/Semisimple_module

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (2)

Related publications

Related people

Related units

Related concepts (23)

Related concepts

Related courses (2)

Study the basics of representation theory of groups and associative algebras.

The students are going to solidify their knowledge of ring and module theory with a major emphasis on commutative algebra and a minor emphasis on homological algebra.

Related courses

Related lectures (6)

Related lectures

Generic direct summands of tensor products for simple algebraic groups and quantum groups at roots of unity

Jonathan Gruber

Let G be either a simple linear algebraic group over an algebraically closed field of characteristic l>0 or a quantum group at an l-th root of unity. The category Rep(G) of finite-dimensional G-modules is non-semisimple. In this thesis, we develop new techniques for studying Krull-Schmidt decompositions of tensor products of G-modules.More specifically, we use minimal complexes of tilting modules to define a tensor ideal of singular G-modules, and we show that, up to singular direct summands, taking tensor products of G-modules respects the decomposition of Rep(G) into linkage classes. In analogy with the classical translation principle, this allows us to reduce questions about tensor products of G-modules in arbitrary l-regular linkage classes to questions about tensor products of G-modules in the principal block of G. We then identify a particular non-singular indecomposable direct summand of the tensor product of two simple G-modules in the principal block (with highest weights in two given l-alcoves), which we call the generic direct summand because it appears generically in Krull-Schmidt decompositions of tensor products of simple G-modules (with highest weights in the given l-alcoves). We initiate the study of generic direct summands, and we use them to prove a necessary condition for the complete reducibility of tensor products of simple G-modules, when G is a simple algebraic group of type A_n.

EPFL2022

Modules d'endo-permutation

Nadia Mazza

If p is an odd prime, the group of endo-trivial modules for a metacyclic p-group is infinite cyclic generated by the Heller translate of the trivial module. The structure of the whole Dade group of endo-permutation modules can be deduced from this. Every known torsion endo-permutation module can be explicitly realized as the source of a simple module, actually for a p-nilpotent group.

Université de Lausanne2003