Editing Semisimple module

{{Short description|Direct sum of irreducible modules}}{{see also|Semisimple algebra}}
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 (mathematics)|ring]] that is a semisimple module over itself is known as an Artinian '''semisimple ring'''.  Some important rings, such as [[group ring]]s of [[finite group]]s over [[field (mathematics)|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 product]]s of [[matrix ring]]s.

For a group-theory analog of the same notion, see ''[[Semisimple representation]]''.

== Definition ==
A [[module (mathematics)|module]] over a (not necessarily commutative) ring is said to be '''semisimple''' (or '''completely reducible''') if it is the [[direct sum of modules|direct sum]] of [[simple module|simple]] (irreducible) submodules.

For a module ''M'', the following are equivalent:
# ''M'' is semisimple; i.e., a direct sum of irreducible modules.
# ''M'' is the sum of its irreducible submodules.
# Every submodule of ''M'' is a [[direct summand]]: for every submodule ''N'' of ''M'', there is a complement ''P'' such that {{nowrap|1=''M'' = ''N'' ⊕ ''P''}}.
For the proof of the equivalences, see ''{{section link|Semisimple representation#Equivalent characterizations}}''.<!--
For <math>3 \Rightarrow 2</math>, the starting idea is to find an irreducible submodule by picking any nonzero <math>x\in M</math> and letting <math>P</math> be a [[maximal submodule]] such that <math>x \notin P</math>.-- by Zorn's lemma? -- It can be shown that the complement of <math>P</math> is irreducible.{{sfn|ps=|Jacobson|1989|p=120}} -->

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 {{nowrap|'''Z'''}} of integers is not a semisimple module over itself, since the submodule {{nowrap|2'''Z'''}} is not a direct summand.

Semisimple is stronger than [[indecomposable module|completely decomposable]],
which is a [[direct sum of modules|direct sum]] of [[indecomposable module|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'', {{nowrap|''F'' ⊗<sub>''K''</sub> ''M''}} is a semisimple module over {{nowrap|''F'' ⊗<sub>''K''</sub> ''A''}}.

== Properties ==
* If ''M'' is semisimple and ''N'' is a [[submodule]], then ''N'' and {{nowrap|''M'' / ''N''}} are also semisimple.
* An arbitrary [[direct sum]] of semisimple modules is semisimple.
* A module ''M'' is [[finitely generated module|finitely generated]] and semisimple if and only if it is Artinian and its [[radical of a module|radical]] is zero.

== Endomorphism rings ==
* A semisimple module ''M'' over a ring ''R'' can also be thought of as a [[ring homomorphism]] from ''R'' into the ring of [[abelian group]] [[endomorphism]]s 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 ring|von Neumann regular]].{{sfn|ps=|Lam|2001|p=62}}

== Semisimple rings ==

A ring is said to be (left-)'''semisimple''' if it is semisimple as a left module over itself.{{sfn|ps=|Sengupta|2012|p=125}} 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
: <math>0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 </math>
there exists {{nowrap|''s'' : ''C'' → ''B''}} such that the composition {{nowrap|''g'' ∘ ''s'' : ''C'' → ''C''}} is the identity. The map ''s'' is known as a section. From this it follows that
: <math>B \cong A \oplus C</math>
or in more exact terms
: <math>B \cong f(A) \oplus s(C).</math>

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

Semisimple rings are of particular interest to algebraists. For example, if the base ring ''R'' is semisimple, then all ''R''-modules would automatically be semisimple. Furthermore, every simple (left) ''R''-module is isomorphic to a minimal left ideal of ''R'', that is, ''R'' is a left [[Kasch ring]].

Semisimple rings are both [[Artinian ring|Artinian]] and [[Noetherian ring|Noetherian]]. From the above properties, a ring is semisimple if and only if it is Artinian and its [[Jacobson radical]] is zero.

If an Artinian semisimple ring contains a field as a [[Center of a ring|central]] [[subring]], it is called a [[semisimple algebra]].

=== Examples ===
* For a [[commutative ring]], the four following properties are equivalent: being a [[semisimple ring]]; being [[Artinian ring|artinian]] and [[reduced ring|reduced]];{{sfn|ps=|Bourbaki|2012|p=133|loc=VIII}} being a [[reduced ring|reduced]] [[Noetherian ring]] of [[Krull dimension]] 0; and being isomorphic to a finite direct product of fields.
* If ''K'' is a field and ''G'' is a finite group of order ''n'', then the [[group ring]] ''K''[''G''] is semisimple if and only if the [[characteristic (algebra)|characteristic]] of ''K'' does not divide ''n''. This is [[Maschke's theorem]], an important result in [[group representation theory]].
* By the [[Wedderburn–Artin theorem]], a unital ring ''R'' is semisimple if and only if it is (isomorphic to) {{nowrap|M<sub>''n''<sub>1</sub></sub>(''D''<sub>1</sub>) × M<sub>''n''<sub>2</sub></sub>(''D''<sub>2</sub>) × ... × M<sub>''n''<sub>''r''</sub></sub>(''D''<sub>''r''</sub>)}}, where each ''D''<sub>''i''</sub> is a [[division ring]] and each ''n''<sub>''i''</sub> is a positive integer, and M<sub>''n''</sub>(''D'') denotes the ring of ''n''-by-''n'' matrices with entries in ''D''.
* An example of a semisimple non-unital ring is M<sub>∞</sub>(''K''), the row-finite, column-finite, infinite matrices over a field ''K''.

=== Simple rings ===
{{main|Simple ring}}

One should beware that despite the terminology, ''not all simple rings are semisimple''. The problem is that the ring may be "too big", that is, not (left/right) Artinian. In fact, if ''R'' is a simple ring with a minimal left/right ideal, then ''R'' is semisimple.

Classic examples of simple, but not semisimple, rings are the [[Weyl algebra]]s, such as the '''Q'''-algebra
: <math> A=\mathbf{Q}{\langle x,y \rangle }/\langle xy-yx-1\rangle\ ,</math>
which is a simple noncommutative [[domain (ring theory)|domain]].  These and many other nice examples are discussed in more detail in several noncommutative ring theory texts, including chapter 3 of Lam's text, in which they are described as nonartinian simple rings.  The [[module theory]] for the Weyl algebras is well studied and differs significantly from that of semisimple rings.

=== Jacobson semisimple ===
{{main|Semiprimitive ring}}
A ring is called ''Jacobson semisimple'' (or ''J-semisimple'' or ''[[semiprimitive ring|semiprimitive]]'') if the intersection of the maximal left ideals is zero, that is, if the [[Jacobson radical]] is zero.  Every ring that is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself.  A J-semisimple ring is semisimple if and only if it is an [[artinian ring]], so semisimple rings are often called ''artinian semisimple rings'' to avoid confusion.

For example, the ring of integers, '''Z''', is J-semisimple, but not artinian semisimple.

== See also ==

* [[Socle (mathematics)|Socle]]
* [[Semisimple algebra]]

== Citations ==
{{reflist}}

== References ==
{{refbegin}}
* {{citation | last1=Bourbaki | first1=Nicolas | year=2012 | title=Algèbre Ch. 8 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-3-540-35315-7 }}
* {{citation | last1=Jacobson | first1=Nathan | year=1989 | author1-link=Nathan Jacobson | title=Basic algebra II | publisher=W. H. Freeman | edition=2nd | isbn=978-0-7167-1933-5 }}
* {{citation | last1=Lam | first1=Tsit-Yuen | year=2001 | title=A First Course in Noncommutative Rings | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-95325-0 | mr=1838439 | volume=131 | doi=10.1007/978-1-4419-8616-0 }}
* {{citation | last1=Lang | first1=Serge | year=2002 | authorlink = Serge Lang| title=Algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | isbn=978-0387953854 }}
* {{citation | last1=Pierce | first1=R.S. | year=1982 | title=Associative Algebras | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | isbn=978-1-4757-0165-4 }}
* {{cite book | last=Sengupta | first=Ambar | year=2012 | title=Representing finite groups: a semisimple introduction | chapter=Induced Representations | pages=235–248 | isbn=9781461412311 | location=New York | doi=10.1007/978-1-4614-1231-1_8 | oclc=769756134 }}
{{refend}}

[[Category:Module theory]]
[[Category:Ring theory]]