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In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every cyclic submodule generated by a non-zero element of ''M'' equals ''M''. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory. In this article, all modules will be assumed to be right unital modules over a ring ''R''. == Examples == Z-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups. These are the cyclic groups of prime order. If ''I'' is a right ideal of ''R'', then ''I'' is simple as a right module if and only if ''I'' is a 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. 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 ''R''/''I'' is simple if and only if ''I'' is a maximal right ideal: If ''M'' is a non-zero proper submodule of ''R''/''I'', then the preimage of ''M'' under the quotient map 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 has a non-zero kernel which is not equal to , and therefore is not simple. Every simple ''R''-module is isomorphic to a quotient ''R''/''m'' where ''m'' is a maximal right ideal of ''R''.〔Herstein, ''Non-commutative Ring Theory'', Lemma 1.1.3〕 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 ''xR'' = ''M'' is equivalent to the surjectivity of the homomorphism 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 and ''G'' is a group, then a group representation of ''G'' is a left module over the group ring ''k()''. The simple ''k()'' modules are also known as irreducible representations. A major aim of representation theory is to understand the irreducible representations of groups. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Simple module」の詳細全文を読む スポンサード リンク
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