HOME

TheInfoList




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 their changes (cal ...
, specifically in
ring theory 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. ...
, 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: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
s over ''R'' that are non-zero and have no non-zero proper
submodule 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. Equivalently, a module ''M'' is simple
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
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 ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...
, 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 as
abelian 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 an ...
s, so a simple Z-module is an abelian group which has no non-zero proper
subgroup 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 ...
s. These are the
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

cyclic group
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 Order, ORDER or Orders may refer to: * Orderliness Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
. If ''I'' is a right
ideal Ideal may refer to: Philosophy * Ideal (ethics) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
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 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 an ...
of ''M'' under the
quotient map In topology 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), ...
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: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
which is not equal to , and therefore is not simple. Every simple ''R''-module is
isomorphic 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 ...

isomorphic
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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
and ''G'' is a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, then a
group representation In the mathematical 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 quantiti ...
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 of
length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...
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 \to N \to M \to M/N \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 :\cdots \subset M_2 \subset M_1 \subset M. In order to prove the fact this way, one needs conditions on this sequence and on the modules ''M''''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.


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'' = 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''·''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.


See also

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


References

{{DEFAULTSORT:Simple Module Module theory Representation theory