Semisimple module

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.

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:

1. M is semisimple; i.e., a direct sum of irreducible modules.
2. M is the sum of its irreducible submodules.
3. Every submodule of M is a direct summand: for every submodule N of M, there is a complement P such that M = N ⊕ P.

For the proof of the equivalences, see Semisimple representation § Equivalent characterizations.

The most basic example of a semisimple module is a module over a field, i.e., a vector space. On the other hand, the ring Z of integers is not a semisimple module over itself, since the submodule 2Z is not a direct summand.

Semisimple is stronger than completely decomposable, which is a direct sum of indecomposable submodules.

Let A be an algebra over a field K. Then a left module M over A is said to be absolutely semisimple if, for any field extension F of K, F ⊗_KM is a semisimple module over F ⊗_KA.

• If M is semisimple and N is a submodule, then N and M / N are also semisimple.
• The direct sum of (possibly infinitely many) semisimple modules is semisimple.
• A module M is finitely generated and semisimple if and only if it is Artinian and its radical is zero.

• A semisimple module M over a ring R can also be thought of as a ring homomorphism from R into the ring of abelian groupendomorphisms of M. The image of this homomorphism is a semiprimitive ring, and every semiprimitive ring is isomorphic to such an image.
• The endomorphism ring of a semisimple module is not only semiprimitive, but also von Neumann regular.^[1]

A ring is said to be (left-)semisimple if it is semisimple as a left module over itself.^[2] Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right distinction is therefore unnecessary, and one can speak of semisimple rings without ambiguity.

A semisimple ring may be characterized in terms of homological algebra: namely, a ring R is semisimple if and only if any short exact sequence of left (or right) R-modules splits. That is, for a short exact sequence

${\displaystyle 0\to A\xrightarrow {f} B\xrightarrow {g} C\to 0}$

there exists s : C → B such that the composition g ∘ s : C → C is the identity. The map s is known as a section. From this it follows that

${\displaystyle B\cong A\oplus C}$

or in more exact terms

${\displaystyle B\cong f(A)\oplus s(C).}$

In particular, any module over a semisimple ring is injective and projective. Since "projective" implies "flat", a semisimple ring is a von Neumann regular ring.

Los anillos semisimples son de particular interés para los algebristas. Por ejemplo, si el anillo base R es semisimple, entonces todos los R -módulos serían automáticamente semisimples. Además, todo R -módulo simple (izquierdo) es isomorfo a un ideal izquierdo mínimo de R , es decir, R es un anillo de Kasch izquierdo .

Los anillos semisimples son artinianos y noetherianos . Según las propiedades anteriores, un anillo es semisimple si y solo si es artiniano y su radical de Jacobson es cero.

Si un anillo semisimple artiniano contiene un cuerpo como subanillo central , se le llama álgebra semisimple .

• Socle
• Semisimple algebra