Módulo semisimple

En matemáticas, concretamente en álgebra abstracta y teoría de módulos, un módulo semisimple o módulo completamente reducible es un tipo de módulo que se expresa como suma directa de submódulos simples. Los anillos que son módulos semisimples sobre sí mismos se llaman anillos semisimples, y su estructura queda determinada por el Teorema de Artin-Wedderburn, que los caracteriza como isomorfos a un producto directo de anillos de matrices sobre anillos de división.

Definición

Un módulo sobre un anillo (no necesariamente conmutativo) se dice semisimple (o completamente reducible) cuando es una suma directa de submódulos simples.

Caracterización

Para un módulo cualquiera $M$ , equivalen:

$M$ es semisimple, suma directa de módulos simples.
$M$ es la suma directa de sus submódulos simples.
Cada submódulo de $M$ es un sumando directo; es decir, para cada submódulo $N$ de $M$ , existe un complemento $P$ tal que $M=N\oplus P$ .

Propiedades

Si $M$ es semisimple y $N$ es un submódulo, entonces $N$ y $M/N$ son también semisimples.
Si cada $M_{i}$ es un módulo semisimple, también lo es $\bigoplus _{i}M_{i}$ .
Un módulo $M$ es finitamente generado y semisimple si y sólo si es artiniano y su radical es cero.

Referencias

Bourbaki, Nicolas (2012), Algèbre Ch. 8 (2nd edición), Berlin, New York: Springer-Verlag, ISBN 978-3-540-35315-7 .
Jacobson, Nathan (1989), Basic algebra II (2nd edición), W. H. Freeman, ISBN 978-0-7167-1933-5 .
Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings (2nd edición), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0, MR 1838439 .
Lang, Serge (2002), Algebra (3rd edición), Berlin, New York: Springer-Verlag, ISBN 978-0387953854 .
R.S. Pierce. Associative Algebras. Graduate Texts in Mathematics vol 88.

Datos: Q1485102

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