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...

reducibility says a finite-dimensional representation of a semisimple compact Lie group is semisimple. A square matrix (in other words a linear operator T :...

are semisimple or can be approximated by semisimple representations. A semisimple module over an algebra over a field is an example of a semisimple representation...

and S. Simple modules are sometimes called irreducible. Semisimple A semisimple module is a direct sum (finite or not) of simple modules. Historically...

simple modules. Over semisimple rings, this is no loss as every module is a semisimple module and so a direct sum of simple modules. Ordinary character...

essential submodule of M, and soc(M) is f.g. If M is a semisimple module (such as soc(N) for any module N), it is f.g. if and only if f.cog. If M is f.g. and...

example, a semisimple module is a module that has a decomposition into simple modules. Given a ring, the types of decomposition of modules over the ring...

completely reducible (or semisimple) if it is isomorphic to a direct sum of irreducible representations (cf. semisimple module). If V is finite-dimensional...

weaker than being semisimple, which is a direct sum of simple modules. A direct sum decomposition of a module into indecomposable modules is called an indecomposable...

Jacobson semisimple if its Jacobson radical is the zero ideal. A ring is semiprimitive if and only if it has a faithful semisimple left module. The semiprimitive...

matrix ring Mn(R), the natural module Rn is projective but is not free when n > 1. Over a semisimple ring, every module is projective, but a nonzero proper...

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any...

are right orders in a semisimple ring. Modules of finite uniform dimension generalize both Artinian modules and Noetherian modules. In the literature, uniform...

algebra. A semisimple module is a direct sum of simple modules. A semisimple ring is a ring that is semisimple as a left module (or right module) over itself...

injective module is injective if and only if the ring is Artinian semisimple (Golan & Head 1991, p. 152); every factor module of every injective module is injective...

a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) semisimple ring R is isomorphic to a product...

of irreducible representations of a complex semisimple Lie algebra. Specifically, although Verma modules themselves are infinite dimensional, quotients...

a semiartinian module. A module is semisimple if and only if soc(M) = M. Rings for which soc(M) = M for all M are precisely semisimple rings. soc(soc(M)) = soc(M)...

In the context of a module M over a ring R, the top of M is the largest semisimple quotient module of M if it exists. For finite-dimensional k-algebras...

In ring theory, a branch of mathematics, a semisimple algebra is an associative Artinian algebra over a field which has trivial Jacobson radical (only...

representations correspond to simple modules. In the module-theoretic language, Maschke's theorem asks: is an arbitrary module semisimple? In this context, the theorem...

the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists...

Any simple module is trivially uniserial, and likewise semisimple modules are serial modules. Many examples of serial rings can be gleaned from the structure...

the R module M has any of the following properties if and only if the S module F(M) does: injective, projective, flat, faithful, simple, semisimple, finitely...

finite-dimensional module over g {\displaystyle {\mathfrak {g}}} is semisimple as a module (i.e., a direct sum of simple modules.) Weyl's theorem implies...

symplectic group Sp(2n). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude Chevalley showed that the classification...

algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings...

Localization of a module Completion (ring theory) Types Simple module, Semisimple module Indecomposable module Artinian module, Noetherian module Homological...

algebraic character is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite-dimensional...

to be (left)-semisimple if it is semisimple as a left module over itself. Surprisingly, a left-semisimple ring is also right-semisimple and vice versa...