Editing Simple module (section)

== Examples ==
'''[[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 [[ideal (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'' &rarr; ''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'' &rarr; ''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.