Simple module: Difference between revisions

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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, 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|date=|publisher=Springer-Verlag|year=1977|isbn=0387901906|series=|___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 ==