Editing Indecomposable module

In [[abstract algebra]], a [[module (mathematics)|module]] is '''indecomposable''' if it is non-zero and cannot be written as a [[direct sum of modules|direct sum]] of two non-zero [[submodule]]s.{{sfn|Jacobson|2009|p=111|ps=none}}{{sfn|Roman|2008|loc=p. 158 §6|ps=none}}

Indecomposable is a weaker notion than [[simple module]] (which is also sometimes called [[irreducibility (mathematics)|irreducible]] module):
simple means "no proper submodule" {{nowrap|''N'' < ''M''}},
while indecomposable "not expressible as {{nowrap|1=''N'' ⊕ ''P'' = ''M''}}".

A direct sum of indecomposables is called '''completely decomposable''';{{fact|date=December 2019}}<!-- this terminology does not appear in the reference (Jacobson) --> this is weaker than being [[semisimple module|semisimple]], which is a direct sum of [[simple module]]s.

A direct sum decomposition of a module into indecomposable modules is called an [[indecomposable decomposition]].

== Motivation ==
In many situations, all modules of interest are completely decomposable; the indecomposable modules can then be thought of as the "basic building blocks", the only objects that need to be studied. This is the case for modules over a
[[field (mathematics)|field]] or [[principal ideal domain|PID]],
and underlies [[Jordan normal form]] of [[linear operator|operators]].

== Examples ==
=== Field ===
Modules over [[field (mathematics)|field]]s are [[vector space]]s.{{sfn|Roman|2008|loc=p. 110 §4|ps=none}} A vector space is indecomposable if and only if its [[dimension (linear algebra)|dimension]] is 1. So every vector space is completely decomposable (indeed, semisimple), with infinitely many summands if the dimension is infinite.{{sfn|Jacobson|2009|loc=p. 111, in comments after Prop. 3.1|ps=none}}

=== Principal ideal domain ===
Finitely-generated modules over [[principal ideal domain]]s (PIDs) 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''/''p''<sup>''n''</sup> for [[prime ideal]]s ''p'' (including {{nowrap|1=''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 {{nowrap|1=''n'' = 1}} (or {{nowrap|1=''p'' = 0}}); for example, the [[cyclic group]] of order 4, '''Z'''/4, is indecomposable but not simple – it has the subgroup 2'''Z'''/4 of order 2, but this does not have a complement.

Over the [[integer]]s '''Z''', modules are [[abelian group]]s. A finitely-generated abelian group is indecomposable if and only if it is [[group isomorphism|isomorphic]] to '''Z''' or to a [[factor group]] of the form '''Z'''/''p''<sup>''n''</sup>'''Z''' for some [[prime number]] ''p'' and some positive integer ''n''. Every [[finitely-generated abelian group]] is a [[Direct sum of groups|direct sum]] of (finitely many) indecomposable abelian groups.

There are, however, other indecomposable abelian groups which are not finitely generated; examples are the [[rational number]]s '''Q''' and the [[Prüfer group|Prüfer ''p''-group]]s '''Z'''(''p''<sup>∞</sup>) for any prime number ''p''.

For a fixed positive integer ''n'', consider the ring ''R'' of ''n''-by-''n'' [[matrix (mathematics)|matrices]] with entries from the [[real number]]s (or from any other field ''K''). Then ''K''<sup>''n''</sup> 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 ''K''<sup>''n''</sup>.

==Facts==
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 (ring theory)#Role in decompositions|idempotent element]] different from 0 and 1.{{sfn|Jacobson|2009|p=111|ps=none}} (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 of a module|length]] is indecomposable if and only if its endomorphism ring is [[local ring|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).{{sfn|Jacobson|2009|p=115|ps=none}}

== Citations ==
{{reflist}}

== References ==
* {{citation
  |last=Jacobson |first=Nathan |author-link=Nathan Jacobson
  |date=2009
  |title=Basic algebra
  |edition=2nd |volume = 2
  |publisher=Dover
  |isbn = 978-0-486-47187-7
  }}
* {{citation
  |title=Advanced linear algebra
  |edition=3rd
  |last=Roman |first=Steven
  |year=2008 |publisher=Springer Science + Business Media |place=New York |isbn=978-0-387-72828-5}}

{{DEFAULTSORT:Indecomposable Module}}
[[Category:Module theory]]