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 ...
, especially in the area of
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
known as
module theory 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 ...
, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A
ring Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as
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 ...
s of
finite group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
s over
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
of characteristic zero, are semisimple rings. An
Artinian ring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the
Artin–Wedderburn theorem 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 ...
, which exhibits these rings as finite
direct product 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 the ...
s of
matrix ring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s. For a group-theory analog of the same notion, see ''
Semisimple representation In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group (mathematics), group or an algebra over a field, algebra that is a direct s ...
''.

Definition

A
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 ...
over a (not necessarily commutative) ring is said to be semisimple (or completely reducible) if it is the
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * Simple (album), ''Simple'' (album), by Andy Yorke, 2008, and its title track * Simple (Florida Georgia Line song), "Simple" (Florida Ge ...
(irreducible) submodules. For a module ''M'', the following are equivalent: # ''M'' is semisimple; i.e., a direct sum of irreducible modules. # ''M'' is the sum of its irreducible submodules. # Every submodule of ''M'' is a
direct summand The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
: for every submodule ''N'' of ''M'', there is a complement ''P'' such that . For the proof of the equivalences, see '. The most basic example of a semisimple module is a module over a field, i.e., a
vector space 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 a ...
. On the other hand, the ring of integers is not a semisimple module over itself, since the submodule is not a direct summand. Semisimple is stronger than completely decomposable, which is a
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of indecomposable submodules. Let ''A'' be an algebra over a field ''K''. Then a left module ''M'' over ''A'' is said to be absolutely semisimple if, for any field extension ''F'' of ''K'', is a semisimple module over .

Properties

* If ''M'' is semisimple and ''N'' is a
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 ...
, then ''N'' and ''M''/''N'' are also semisimple. * An arbitrary
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of semisimple modules is semisimple. * A module ''M'' is finitely generated and semisimple if and only if it is Artinian and its
is zero.

Endomorphism rings

* A semisimple module ''M'' over a ring ''R'' can also be thought of as a
ring homomorphism 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 an ...
from ''R'' into the ring of
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 ...
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group ...
s of ''M''. The image of this homomorphism is a
semiprimitive ring In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more general than a semisimple ring, but where simple modules still provide enough information about ...
, and every semiprimitive ring is isomorphic to such an image. * The
endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a Point ...
of a semisimple module is not only semiprimitive, but also von Neumann regular, .

Semisimple rings

A ring is said to be (left)-semisimple if it is semisimple as a left module over itself. Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right distinction is therefore unnecessary, and one can speak of semisimple rings without ambiguity. A semisimple ring may be characterized in terms of homological algebra: namely, a ring ''R'' is semisimple if and only if any short exact sequence of left (or right) ''R''-modules splits. That is for a short exact sequence :$0 \to A \xrightarrow B \xrightarrow C \to 0$ there exists such that the composition is the identity. The map ''s'' is known as a section. From this is follows that :$B \cong A \oplus C$ or in more exact terms :$B \cong f\left(A\right) \oplus s\left(C\right)$ In particular, any module over a semisimple ring is
injective 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 ...

and . Since "projective" implies "flat", a semisimple ring is a
von Neumann regular ring In mathematics, a von Neumann regular ring is a ring (mathematics), ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inv ...
. Semisimple rings are of particular interest to algebraists. For example, if the base ring ''R'' is semisimple, then all ''R''-modules would automatically be semisimple. Furthermore, every simple (left) ''R''-module is isomorphic to a minimal left ideal of ''R'', that is, ''R'' is a left Kasch ring. Semisimple rings are both Artinian and
Noetherian In mathematics, the adjective Noetherian is used to describe Category_theory#Categories.2C_objects.2C_and_morphisms, objects that satisfy an ascending chain condition, ascending or descending chain condition on certain kinds of subobjects, meaning t ...
. From the above properties, a ring is semisimple if and only if it is Artinian and its
Jacobson radical 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 the ...
is zero. If an Artinian semisimple ring contains a field as a
central Central is an adjective In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign languag ...
subring 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 ...
, it is called a
semisimple algebra In ring theory, a branch of mathematics, a semisimple algebra is an associative algebra, associative artinian ring, artinian algebra over a field (mathematics), field which has trivial Jacobson radical (only the zero element of the algebra is in t ...
.

Examples

* A commutative semisimple ring is a finite direct product of fields. A commutative ring is semisimple if and only if it is artinian and reduced. * If ''K'' is a field and ''G'' is a finite group of order ''n'', then 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''is semisimple if and only if the
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
of ''K'' does not divide ''n''. This is
Maschke's theorem 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 th ...
, an important result in
group representation theory In the mathematics, mathematical field of representation theory, group representations describe abstract group (mathematics), groups in terms of bijective linear transformations (i.e. automorphisms) of vector spaces; in particular, they can be ...
. * By the
Artin–Wedderburn theorem 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 ...
, a unital Artinian ring ''R'' is semisimple if and only if it is (isomorphic to) , where each ''D''''i'' is a
division 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. In ...
and each ''n''''i'' is a positive integer, and M''n''(''D'') denotes the ring of ''n''-by-''n'' matrices with entries in ''D''. * An example of a semisimple non-unital ring is M(''K''), the row-finite, column-finite, infinite matrices over a field ''K''.

Simple rings

One should beware that despite the terminology, ''not all simple rings are semisimple''. The problem is that the ring may be "too big", that is, not (left/right) Artinian. In fact, if ''R'' is a simple ring with a minimal left/right ideal, then ''R'' is semisimple. Classic examples of simple, but not semisimple, rings are the
Weyl algebra In abstract algebra, the Weyl algebra is the ring (mathematics), ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X ...
s, such as the $\mathbb$-algebra :$A=\mathbb/\langle xy-yx-1\rangle\ ,$ which is a simple noncommutative
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
. These and many other nice examples are discussed in more detail in several noncommutative ring theory texts, including chapter 3 of Lam's text, in which they are described as nonartinian simple rings. The
module theory 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 ...
for the Weyl algebras is well studied and differs significantly from that of semisimple rings.

Jacobson semisimple

A ring is called ''Jacobson semisimple'' (or ''J-semisimple'' or '' semiprimitive'') if the intersection of the maximal left ideals is zero, that is, if the
Jacobson radical 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 the ...
is zero. Every ring that is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself. A J-semisimple ring is semisimple if and only if it is an
artinian ring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
, so semisimple rings are often called ''artinian semisimple rings'' to avoid confusion. For example, the ring of integers, Z, is J-semisimple, but not artinian semisimple.