Central Idempotent
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In ring theory, a branch of
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 ...
, an idempotent element or simply idempotent of a
ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
is an element such that . That is, the element is
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
under the ring's multiplication. Inductively then, one can also conclude that for any positive
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
. For example, an idempotent element of a
matrix ring In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication. The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'') (alternat ...
is precisely an
idempotent matrix In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix A is idempotent if and only if A^2 = A. For this product A^2 to be defined, A must necessarily be a square matrix. Viewed thi ...
. For general rings, elements idempotent under multiplication are involved in decompositions of
modules Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computer science and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components ...
, and connected to
homological Homology, homologous, homologation or homological may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor *Sequence homology, biological homology between DNA, RNA, ...
properties of the ring. In
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
, the main objects of study are rings in which all elements are idempotent under both addition and multiplication.


Examples


Quotients of Z

One may consider the ring of integers modulo , where is
square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. ...
. By the
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
, this ring factors into the
product of rings In mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. It is a direct product in the ...
of integers modulo , where is
prime A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only ways ...
. Now each of these factors 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 grass ...
, so it is clear that the factors' only idempotents will be and . That is, each factor has two idempotents. So if there are factors, there will be idempotents. We can check this for the integers , . Since has two prime factors ( and ) it should have idempotents. : : : : : : From these computations, , , , and are idempotents of this ring, while and are not. This also demonstrates the decomposition properties described below: because , there is a ring decomposition . In the multiplicative identity is and in the multiplicative identity is .


Quotient of polynomial ring

Given a ring and an element such that , the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
: has the idempotent . For example, this could be applied to , or any
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
.


Idempotents in the ring of split-quaternions

There is a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
of idempotents in the ring of
split-quaternion In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers. After introduction in t ...
s. Split quaternions have the structure of a real algebra, so elements can be written ''w'' + ''x''i + ''y''j + ''z''k over a
basis Basis is a term used in mathematics, finance, science, and other contexts to refer to foundational concepts, valuation measures, or organizational names; here, it may refer to: Finance and accounting * Adjusted basis, the net cost of an asse ...
, with j2 = k2 = +1. For any θ, :s = j \cos \theta + k \sin \theta satisfies s2 = +1 since j and k satisfy the
anticommutative property In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
. Now :(\frac)^2 = \frac = \frac, the idempotent property. The element ''s'' is called a
hyperbolic unit In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1, where j \neq \pm 1. A split-complex number has two real number components and , and is written z=x+yj ...
and so far, the i-coordinate has been taken as zero. When this coordinate is non-zero, then there is a
hyperboloid of one sheet In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface (mathematics), surface generated by rotating a hyperbola around one of its Hyperbola#Equation, principal axes. A hyperboloid is the surface obtained ...
of hyperbolic units in split-quaternions. The same equality shows the idempotent property of \frac where ''s'' is on the hyperboloid.


Types of ring idempotents

A partial list of important types of idempotents includes: * Two idempotents and are called orthogonal if . If is idempotent in the ring (with
unity Unity is the state of being as one (either literally or figuratively). It may also refer to: Buildings * Unity Building, Oregon, Illinois, US; a historic building * Unity Building (Chicago), Illinois, US; a skyscraper * Unity Buildings, Liverpoo ...
), then so is ; moreover, and are orthogonal. * An idempotent in is called a central idempotent if for all in , that is, if is in the
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
of . * A trivial idempotent refers to either of the elements and , which are always idempotent. * A primitive idempotent of a ring is a nonzero idempotent such that is indecomposable as a right -module; that is, such that is not 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 two nonzero
submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since t ...
s. Equivalently, is a primitive idempotent if it cannot be written as , where and are nonzero orthogonal idempotents in . * A local idempotent is an idempotent such that 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 ...
. This implies that is directly indecomposable, so local idempotents are also primitive. * A right irreducible idempotent is an idempotent for which is a
simple module In mathematics, 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 every ...
. By
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 ...
, 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 ...
, and hence is a local ring, so right (and left) irreducible idempotents are local. * A centrally primitive idempotent is a central idempotent that cannot be written as the sum of two nonzero orthogonal central idempotents. * An idempotent in the quotient ring is said to lift modulo if there is an idempotent in such that . * An idempotent of is called a full idempotent if . * A separability idempotent; see ''
Separable algebra In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension. Definition and first properties A homomorphism of (unital, but not necessarily ...
''. Any non-trivial idempotent is a
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
(because with neither nor being zero, where ). This shows that
integral domain In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...
s and
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 ...
s do not have such idempotents.
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 ...
s also do not have such idempotents, but for a different reason. The only idempotent contained in the
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...
of a ring is .


Rings characterized by idempotents

* A ring in which ''all'' elements are idempotent is called a
Boolean ring In mathematics, a Boolean ring is a ring for which for all in , that is, a ring that consists of only idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean algebra, with ring multiplicat ...
. Some authors use the term "idempotent ring" for this type of ring. In such a ring, multiplication is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
and every element is its own
additive inverse In mathematics, the additive inverse of an element , denoted , is the element that when added to , yields the additive identity, 0 (zero). In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero el ...
. * A ring is
semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
if and only if every right (or every left)
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
is generated by an idempotent. * A ring is
von Neumann regular In mathematics, a von Neumann regular ring is a 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 inverse" of the elemen ...
if and only if every finitely generated right (or every finitely generated left) ideal is generated by an idempotent. * A ring for which the annihilator every subset of is generated by an idempotent is called a Baer ring. If the condition only holds for all
singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...
subsets of , then the ring is a right Rickart ring. Both of these types of rings are interesting even when they lack a multiplicative identity. * A ring in which all idempotents are central is called an abelian ring. Such rings need not be commutative. * A ring is directly irreducible if and only if and are the only central idempotents. * A ring can be written as with each a local idempotent if and only if is a
semiperfect ring In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring over which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there ...
. * A ring is called an
SBI ring In abstract algebra, algebra, an SBI ring is a ring (mathematics), ring ''R'' (with identity) such that every idempotent (ring theory), idempotent of ''R'' modulo (jargon), modulo the Jacobson radical can be lift (mathematics), lifted to ''R''. The ...
or Lift/rad ring if all idempotents of lift modulo the
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...
. * A ring satisfies the
ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings. These conditions p ...
on right direct summands if and only if the ring satisfies the
descending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings. These conditions played an important r ...
on left direct summands if and only if every set of pairwise orthogonal idempotents is finite. * If is idempotent in the ring , then is again a ring, with multiplicative identity . The ring is often referred to as a corner ring of . The corner ring arises naturally since the
ring of endomorphisms 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 a p ...
.


Role in decompositions

The idempotents of have an important connection to decomposition of -
modules Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computer science and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components ...
. If is an -module and is its
ring of endomorphisms 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 a p ...
, then if and only if there is a unique idempotent in such that and . Clearly then, is directly indecomposable if and only if and are the only idempotents in . In the case when (assumed unital), the endomorphism ring , where each
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 g ...
arises as left multiplication by a fixed ring element. With this modification of notation, as right modules if and only if there exists a unique idempotent such that and . Thus every direct summand of is generated by an idempotent. If is a central idempotent, then the corner ring is a ring with multiplicative identity . Just as idempotents determine the direct decompositions of as a module, the central idempotents of determine the decompositions of as 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 rings. If is the direct sum of the rings , ..., , then the identity elements of the rings are central idempotents in , pairwise orthogonal, and their sum is . Conversely, given central idempotents , ..., in that are pairwise orthogonal and have sum , then is the direct sum of the rings , ..., . So in particular, every central idempotent in gives rise to a decomposition of as a direct sum of the corner rings and . As a result, a ring is directly indecomposable as a ring if and only if the identity is centrally primitive. Working inductively, one can attempt to decompose into a sum of centrally primitive elements. If is centrally primitive, we are done. If not, it is a sum of central orthogonal idempotents, which in turn are primitive or sums of more central idempotents, and so on. The problem that may occur is that this may continue without end, producing an infinite family of central orthogonal idempotents. The condition "'' does not contain infinite sets of central orthogonal idempotents''" is a type of finiteness condition on the ring. It can be achieved in many ways, such as requiring the ring to be right
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
. If a decomposition exists with each a centrally primitive idempotent, then is a direct sum of the corner rings , each of which is ring irreducible. For
associative algebra In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
s or
Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan a ...
s over a field, the
Peirce decomposition Peirce may refer to: * Charles Sanders Peirce (1839–1914), American philosopher, founder of pragmatism Schools * Peirce College, Philadelphia, formerly known as Peirce College of Business, Peirce Junior College and Peirce School of Business Admi ...
is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements.


Relation with involutions

If is an idempotent of the endomorphism ring , then the endomorphism is an -module
involution Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...
of . That is, is an -
module homomorphism In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a ring ''R'', then a function f: M \to N is called an ''R''-''module homomorphism'' or an ' ...
such that is the identity endomorphism of . An idempotent element of and its associated involution gives rise to two involutions of the module , depending on viewing as a left or right module. If represents an arbitrary element of , can be viewed as a right -module homomorphism so that , or can also be viewed as a left -module homomorphism , where . This process can be reversed if is an
invertible element In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
of : if is an involution, then and are orthogonal idempotents, corresponding to and . Thus for a ring in which is invertible, the idempotent elements correspond to involutions in a one-to-one manner.


Category of ''R''-modules

Lifting idempotents also has major consequences for the category of -modules. All idempotents lift modulo if and only if every direct summand of has a
projective cover In the branch of abstract mathematics called category theory, a projective cover of an object ''X'' is in a sense the best approximation of ''X'' by a projective object ''P''. Projective covers are the dual of injective envelopes. Definition ...
as an -module. Idempotents always lift modulo
nil ideal In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if all of its elements is nilpotent, i.e for each a \in I exists natural number ''n'' for which a^n = 0. If all elements of a ring ...
s and rings for which is -adically complete. Lifting is most important when , the
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...
of . Yet another characterization of
semiperfect ring In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring over which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there ...
s is that they are
semilocal ring In mathematics, a semi-local ring is a ring for which ''R''/J(''R'') is a semisimple ring, where J(''R'') is the Jacobson radical of ''R''. The above definition is satisfied if ''R'' has a finite number of maximal right ideals (and finite number ...
s whose idempotents lift modulo .


Lattice of idempotents

One may define a
partial order In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable ...
on the idempotents of a ring as follows: if and are idempotents, we write if and only if . With respect to this order, is the smallest and the largest idempotent. For orthogonal idempotents and , is also idempotent, and we have and . The
atoms Atoms are the basic particles of the chemical elements. An atom consists of a nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished from each other ...
of this partial order are precisely the primitive idempotents. When the above partial order is restricted to the central idempotents of , a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an or ...
structure, or even a
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
structure, can be given. For two central idempotents and , the
complement Complement may refer to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class collections into complementary sets * Complementary color, in the visu ...
is given by : , the
meet Meet may refer to: People with the name * Janek Meet (born 1974), Estonian footballer * Meet Mukhi (born 2005), Indian child actor Arts, entertainment, and media * ''Meet'' (TV series), an Australian television series * '' Meet: Badlegi Duniya K ...
is given by : . and the
join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two topo ...
is given by : The ordering now becomes simply if and only if , and the join and meet satisfy and . It is shown in that if is
von Neumann regular In mathematics, a von Neumann regular ring is a 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 inverse" of the elemen ...
and right self-injective, then the lattice is a
complete lattice In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum ( join) and an infimum ( meet). A conditionally complete lattice satisfies at least one of these properties for bounded subsets. For compariso ...
.


Notes


Citations


References

*
idempotent
at
FOLDOC The Free On-line Dictionary of Computing (FOLDOC) is an online, searchable, encyclopedic dictionary of computing subjects. History FOLDOC was founded in 1985 by Denis Howe and was hosted by Imperial College London. In May 2015, the site was up ...
* * * * * * {{refend Ring theory