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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
, and they are analogous to the simple groups in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
. 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
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
s. 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 the quotient module ''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 In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
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, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
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 In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
of ''G'' is a left module over the
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...
''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 Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
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. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
1; this is a reformulation of the definition. Every simple module is indecomposable, but the converse is in general not true. Every simple module is 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 In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
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 An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
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 In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in ...
of any simple module is a
division ring In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...
. This result is known as
Schur's lemma In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a gro ...
. 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 In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
:0 \to N \to M \to M/N \to 0. A common approach to 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
indecomposable module In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. Indecomposable is a weaker notion than simple module (which is also sometimes called irreducible module): simple ...
is a
local ring In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...
, so that the strong Krull–Schmidt theorem holds and the 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 In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a group homomorp ...
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 The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of simple modules. Ordinary character theory provides better arithmetic control, and uses simple C''G'' modules to understand the structure of
finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
s ''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 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 simple ring is isomorphic to a full matrix ring of ''n''-by-''n'' matrices over a
division ring In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...
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