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_{''i''} /''M''_{''i'' + 1}. One particularly useful condition is that the length of the sequence is finite and each quotient module ''M''_{''i''} /''M''_{''i'' + 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 rings, 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 groups ''G''. Modular representation theory uses Brauer characters 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.

_{''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''·''A'' = ''x''·''r'' for all ''x'' in ''X''.Isaacs, Theorem 13.14, p. 185
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 Artinian ring, Artinian 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.

mathematics
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, specifically in ring theory
In algebra
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, the simple modules over a ring ''R'' are the (left or right) module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
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* Modula ...

s over ''R'' that are non-zero and have no non-zero proper submodule
In mathematics
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s. Equivalently, a module ''M'' is simple if and only if
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every cyclic submodule generated by a element of ''M'' equals ''M''. Simple modules form building blocks for the modules of finite length
Length is a measure of distance
Distance is a numerical measurement
'
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, and they are analogous to the simple group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s in group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

.
In this article, all modules will be assumed to be right unital modules over a ring ''R''.
Examples

Z-modules are the same asabelian group
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s, so a simple Z-module is an abelian group which has no non-zero proper subgroup
In group theory
In mathematics
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s. These are the cyclic group
In group theory
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s of prime
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

order
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.
If ''I'' is a right ideal
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An ideal is a principle
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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 the quotient module In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...

''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
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of ''M'' under the quotient map
In topology
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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
Kernel may refer to:
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which is not equal to , and therefore is not simple.
Every simple ''R''-module is isomorphic
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to a quotient ''R''/''m'' where ''m'' is a maximal right ideal of ''R''. 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
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

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
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and ''G'' is a group
A group is a number
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, then a group representation
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of ''G'' is a left module
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra. A module over a ring (mathematics), ring is a generalization of the notion of vector space over a Field (mathematics), field, wherein the correspond ...

over the group ring
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

''k'' 'G''(for details, see the main page on this relationship). 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

The simple modules are precisely the modules oflength
Length is a measure of distance
Distance is a numerical measurement
'
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1; this is a reformulation of the definition.
Every simple module is indecomposable module, indecomposable, but the converse is in general not true.
Every simple module is cyclic module, cyclic, that is it is generated by one element.
Not every module has a simple submodule; consider for instance the Z-module Z in light of the first example above.
Let ''M'' and ''N'' be (left or right) modules over the same ring, and let 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 ''M'' = ''N'', 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 Q is not simple, but its endomorphism ring is isomorphic to the field Q.
Simple modules and composition series

If ''M'' is a module which has a non-zero proper submodule ''N'', then there is a short exact sequence :$0\; \backslash to\; N\; \backslash to\; M\; \backslash to\; M/N\; \backslash to\; 0.$ 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 :$\backslash cdots\; \backslash subset\; M\_2\; \backslash subset\; M\_1\; \backslash subset\; M.$ In order to prove the fact this way, one needs conditions on this sequence and on the modules ''M''The 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 let ''D'' = EndSee also

* Semisimple modules are modules that can be written as a sum of simple submodules * Irreducible ideal * Irreducible representationReferences

{{DEFAULTSORT:Simple Module Module theory Representation theory