Simple module: Difference between revisions

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In [[mathematics]], specifically in [[ring theory]], the '''simple modules''' over a [[Ringring (mathematics)|ring]] ''R'' are the (left or right) [[module (mathematics)|module]]s over ''R'' that are [[Zero_element#Zero_modulezero module|non-zero]] and have no non-zero proper [[submodule]]s. Equivalently, a module ''M'' is simple [[if and only if]] every [[cyclic module|cyclic submodule]] generated by a {{nowrap|non-zero}} element of ''M'' equals ''M''. Simple modules form building blocks for the modules of finite [[length of a module|length]], and they are analogous to the [[simple group]]s in [[group theory]].

In this article, all modules will be assumed to be right [[unital module]]s over a ring ''R''.

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'''[[Integer|Z]]'''-modules are the same as [[abelian group]]s, so a simple '''Z'''-module is an abelian group which has no non-zero proper [[subgroup]]s. These are the [[cyclic group]]s of [[prime number|prime]] [[order (group theory)|order]].

If ''I'' is a right [[Idealideal (ring theory)|ideal]] of ''R'', then ''I'' is simple as a right module if and only if ''I'' is a [[minimal ideal|minimal]] non-zero right ideal: If ''M'' is a non-zero proper submodule of ''I'', then it is also a right ideal, so ''I'' is not minimal. [[Converse (logic)|Conversely]], if ''I'' is not minimal, then there is a non-zero right ideal ''J'' properly contained in ''I''. ''J'' is a right submodule of ''I'', so ''I'' is not simple.

If ''I'' is a right ideal of ''R'', then the [[quotient module]] ''R''/''I'' is simple if and only if ''I'' is a [[maximal ideal|maximal]] right ideal: If ''M'' is a non-zero proper submodule of ''R''/''I'', then the [[preimage]] of ''M'' under the [[Quotient module|quotient map]] {{nowrap|''R'' → ''R''/''I''}} is a right ideal which is not equal to ''R'' and which properly contains ''I''. Therefore, ''I'' is not maximal. Conversely, if ''I'' is not maximal, then there is a right ideal ''J'' properly containing ''I''. The quotient map {{nowrap|''R''/''I'' → ''R''/''J''}} has a non-zero [[Kernel (algebra)|kernel]] which is not equal to {{nowrap|''R''/''I''}}, and therefore {{nowrap|''R''/''I''}} is not simple.

Every simple ''R''-module is [[Module_homomorphism#Terminology|isomorphic]] to a quotient ''R''/''m'' where ''m'' is a [[maximal ideal|maximal]] right ideal]] of ''R''.<ref>Herstein, ''Non-commutative Ring Theory'', Lemma 1.1.3</ref> By the above paragraph, any quotient ''R''/''m'' is a simple module. Conversely, suppose that ''M'' is a simple ''R''-module. Then, for any non-zero element ''x'' of ''M'', the cyclic submodule ''xR'' must equal ''M''. Fix such an ''x''. The statement that {{nowrap begin}}''xR'' = ''M''{{nowrap end}} is equivalent to the [[Surjective|surjectivity]] of the [[Module homomorphism|homomorphism]] {{nowrap|''R'' &rarr; ''M''}} that sends ''r'' to ''xr''. The kernel of this homomorphism is a right ideal ''I'' of ''R'', and a standard theorem states that ''M'' is isomorphic to ''R''/''I''. By the above paragraph, we find that ''I'' is a maximal right ideal. Therefore, ''M'' is isomorphic to a quotient of ''R'' by a maximal right ideal.

If ''k'' is a [[field (mathematics)|field]] and ''G'' is a [[group (mathematics)|group]], then a [[group representation]] of ''G'' is a [[left module]] over the [[group ring]] ''k''[''G]''] (for details, see the [[Representation theory of finite groups#Representations.2C modules and the convolution algebra|main page on this relationship]]).<ref>{{Cite book|url=https://archive.org/details/linearrepresenta1977serr/page/47|title=Linear Representations of Finite Groups|last=Serre|first=Jean-Pierre|publisher=Springer-Verlag|year=1977|isbn=0387901906|___location=New York|pages=[https://archive.org/details/linearrepresenta1977serr/page/47 47]|issn=0072-5285|oclc=2202385}}</ref> The simple ''k''[''G]'' ]-modules are also known as '''irreducible''' representations. A major aim of [[representation theory]] is to understand the irreducible representations of groups.

== Basic properties of simple modules ==

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Let ''M'' and ''N'' be (left or right) modules over the same ring, and let {{nowrap|''f'' : ''M'' → ''N''}} be a module homomorphism. If ''M'' is simple, then ''f'' is either the zero homomorphism or [[injective]] because the kernel of ''f'' is a submodule of ''M''. If ''N'' is simple, then ''f'' is either the zero homomorphism or surjective because the [[image (mathematics)|image]] of ''f'' is a submodule of ''N''. If {{nowrap begin}}''M'' = ''N''{{nowrap end}}, then ''f'' is an [[endomorphism]] of ''M'', and if ''M'' is simple, then the prior two statements imply that ''f'' is either the zero homomorphism or an isomorphism. Consequently, the [[endomorphism ring]] of any simple module is a [[division ring]]. This result is known as '''[[Schur's lemma]]'''.

The converse of Schur's lemma is not true in general. For example, the '''Z'''-module '''[[rational number|Q]]''' is not simple, but its endomorphism ring is isomorphic to the field '''Q'''.

== Simple modules and composition series ==

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If ''M'' is a module which has a non-zero proper submodule ''N'', then there is a [[short exact sequence]]

:<math>0 \to N \to M \to M/N \to 0.</math>

A common approach to [[mathematical proof|proving]] a fact about ''M'' is to show that the fact is true for the center term of a short exact sequence when it is true for the left and right terms, then to prove the fact for ''N'' and ''M''/''N''. If ''N'' has a non-zero proper submodule, then this process can be repeated. This produces a chain of submodules

:<math>\cdots \subset M_2 \subset M_1 \subset M.</math>

In order to prove the fact this way, one needs conditions on this sequence and on the modules ''M''_''i''&thinsp;/''M''_{''i'' &thinsp;+ &hairsp;1}. One particularly useful condition is that the [[length of a module|length]] of the sequence is finite and each quotient module ''M''_''i''&thinsp;/''M''_{''i'' &thinsp;+ &hairsp;1} is simple. In this case the sequence is called a '''composition series''' for ''M''. In order to prove a statement inductively using composition series, the statement is first proved for simple modules, which form the base case of the induction, and then the statement is proved to remain true under an extension of a module by a simple module. For example, the [[Fitting lemma]] shows that the endomorphism ring of a [[finite length module|finite length]] [[indecomposable module]] is a [[local ring]], so that the strong [[Krull–Schmidt theorem]] holds and the [[category (mathematics)|category]] of finite length modules is a [[Krull-Schmidt category]].

The [[Jordan–Hölder theorem]] and the [[Schreier refinement theorem]] describe the relationships amongst all composition series of a single module. The [[Grothendieck group]] ignores the order in a composition series and views every finite length module as a formal sum of simple modules. Over [[semisimple ring]]s, this is no loss as every module is a [[semisimple module]] and so a [[direct sum of modules|direct sum]] of simple modules. [[Ordinary character theory]] provides better arithmetic control, and uses simple '''C'''''G'' modules to understand the structure of [[finite group]]s ''G''. [[Modular representation theory]] uses [[Brauer character]]s to view modules as formal sums of simple modules, but is also interested in how those simple modules are joined together within composition series. This is formalized by studying the [[Ext functor]] and describing the [[module category]] in various ways including [[quiver (mathematics)|quivers]] (whose nodes are the simple modules and whose edges are composition series of non-semisimple modules of length 2) and [[Auslander–Reiten theory]] where the associated graph has a vertex for every indecomposable module.

== The Jacobson density theorem ==

{{main|Jacobson density theorem}}

An important advance in the theory of simple modules was the [[Jacobson density theorem]]. The Jacobson density theorem states:

:Let ''U'' be a simple right ''R''-module and writelet ''D'' = End_''R''(''U''). Let ''A'' be any ''D''-linear operator on ''U'' and let ''X'' be a finite ''D''-linearly independent subset of ''U''. Then there exists an element ''r'' of ''R''' such that ''x''&middot;''A'' = ''x''&middot;''r'' for all ''x'' in ''X''.<ref>Isaacs, Theorem 13.14, p. 185</ref>

In particular, any [[primitive ring]] may be viewed as (that is, isomorphic to) a ring of ''D''-linear operators on some ''D''-space.

A consequence of the Jacobson density theorem is Wedderburn's theorem; namely that any right [[artinianArtinian ring|artinianArtinian]] [[simple ring]] is isomorphic to a full [[matrix ring]] of ''n''-by-''n'' matrices over a [[division ring]] for some ''n''. This can also be established as a [[corollary]] of the [[Artin–Wedderburn theorem]].

==See also==