Utente:Grasso Luigi/sandbox4/Modulo incomponibile

In algebra astratta, un modulo dicesi incomponibile se non è nullo e non può essere posto nella somma diretta di due o più sottomoduli non nulli.

Modulo incomponibile è un concetto più debole di semplice (a volte anche detto modulo irriducibile): semplice per dire sottomodulo non proprio N < M, mentre incomponibile vuole dire M non esprimibile come

$M\ =\ \bigoplus _{i=1}^{n}S_{i}\qquad S_{i}\leq M.$

A direct sum of indecomposables is called completely decomposable; this is weaker than being semisimple, which is a direct sum of simple modules.

La decomposizione di un modulo in una somma diretta di moduli incomponibili viene detta decomposizione incomponibile.

$M\ =\ \bigoplus _{i=1}^{n}MI_{i}$

Motivazione

In molte situazioni tutti i moduli di interesse sono completamente scomponibili; i moduli incomponibili possono quindi essere pensati come i mattoni fondamentali, gli unici oggetti che necessitano di essere studiati. Questo è il caso dei moduli su un campo o PID, in particolare la forma canonica di Jordan di un operatore lineare.

Esempi

Campo

I moduli su un campo sono detti spazi vettoriali. Uno spazio vettoriale è incomponibile se e solo se la sua dimensione vale 1. Quindi ogni spazio vettoriale è completamente scomponibile (anzi, semisemplice), con infiniti addendi se la dimensione è infinita.

Dominio ad ideali principali

Finitely-generated modules over PID are classified by the structure theorem for finitely generated modules over a principal ideal domain: the primary decomposition is a decomposition into indecomposable modules, so every finitely-generated module over a PID is completely decomposable.

Explicitly, the modules of the form R/pⁿ for prime ideals p (including p = 0, which yields R) are indecomposable. Every finitely-generated R-module is a direct sum of these. Note that this is simple if and only if n = 1 (or p = 0); for example, the cyclic group of order 4, Z/4, is indecomposable but not simple – it has the subgroup 2Z/4 of order 2, but this does not have a complement.

Over the integers Z, modules are abelian groups. A finitely-generated abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form Z/pⁿZ for some prime number p and some positive integer n. Every finitely-generated abelian group is a direct sum of (finitely many) indecomposable abelian groups.

There are, however, other indecomposable abelian groups which are not finitely generated; examples are the rational numbers Q and the Prüfer p-groups Z(p^∞) for any prime number p.

For a fixed positive integer n, consider the ring R of n-by-n matrices with entries from the real numbers (or from any other field K). Then Kⁿ is a left R-module (the scalar multiplication is matrix multiplication). This is up to isomorphism the only indecomposable module over R. Every left R-module is a direct sum of (finitely or infinitely many) copies of this module Kⁿ.

Proprietà

Every simple module is indecomposable. The converse is not true in general, as is shown by the second example above.

By looking at the endomorphism ring of a module, one can tell whether the module is indecomposable: if and only if the endomorphism ring does not contain an idempotent element different from 0 and 1. Template:Sfn (If f is such an idempotent endomorphism of M, then M is the direct sum of ker(f) and im(f).)

A module of finite length is indecomposable if and only if its endomorphism ring is local. Still more information about endomorphisms of finite-length indecomposables is provided by the Fitting lemma.

In the finite-length situation, decomposition into indecomposables is particularly useful, because of the Krull–Schmidt theorem: every finite-length module can be written as a direct sum of finitely many indecomposable modules, and this decomposition is essentially unique (meaning that if you have a different decomposition into indecomposables, then the summands of the first decomposition can be paired off with the summands of the second decomposition so that the members of each pair are isomorphic).Template:Sfn

Bibliografia

• (EN) Nathan Jacobson, Basic algebra, vol. 2, 2ª ed., Dover, 2009, ISBN 978-0-486-47187-7.
• (EN) Roman Steven, Advanced linear algebra, 3ª ed., New York, Springer Science + Business Media, 2008, ISBN 978-0-387-72828-5.