Editing Idempotent (ring theory)

{{Short description|In mathematics, element that equals its square}}
In [[ring theory]], a branch of [[mathematics]], an '''idempotent element''' or simply '''idempotent''' of a [[ring (mathematics)|ring]] is an element {{math|''a''}} such that {{math|1=''a''<sup>2</sup> = ''a''}}.{{sfn|ps=none|Hazewinkel|Gubareni|Kirichenko|2004|p=2}}{{efn|Idempotent and [[nilpotent]] were introduced by [[Benjamin Peirce]] in 1870.}} That is, the element is [[idempotent]] under the ring's multiplication. [[Mathematical induction|Inductively]] then, one can also conclude that {{math|1=''a'' = ''a''<sup>2</sup> = ''a''<sup>3</sup> = ''a''<sup>4</sup> = ... = ''a''<sup>''n''</sup>}} for any positive [[integer]] {{math|''n''}}. For example, an idempotent element of a [[matrix ring]] is precisely an [[idempotent matrix]].

For general rings, elements idempotent under multiplication are involved in decompositions of [[module (mathematics)|modules]], and connected to [[homological algebra|homological]] properties of the ring. In [[Boolean algebra]], 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 n|ring of integers modulo {{math|''n''}}]], where {{math|''n''}} is [[square-free integer|square-free]]. By the [[Chinese remainder theorem]], this ring factors into the [[product of rings]] of integers modulo&nbsp;{{math|''p''}}, where {{math|''p''}} is [[prime number|prime]]. Now each of these factors is a [[field (mathematics)|field]], so it is clear that the factors' only idempotents will be {{math|0}} and {{math|1}}. That is, each factor has two idempotents. So if there are {{math|''m''}} factors, there will be {{math|2<sup>''m''</sup>}} idempotents.

We can check this for the integers {{math|mod&nbsp;6}}, {{math|1=''R'' = '''Z''' / 6'''Z'''}}.  Since {{math|6}} has two prime factors ({{math|2}} and {{math|3}}) it should have {{math|2<sup>2</sup>}} idempotents.
: {{math|0<sup>2</sup> ≡ 0 ≡ 0 (mod 6)}}
: {{math|1<sup>2</sup> ≡ 1 ≡ 1 (mod 6)}}
: {{math|2<sup>2</sup> ≡ 4 ≡ 4 (mod 6)}}
: {{math|3<sup>2</sup> ≡ 9 ≡ 3  (mod 6)}}
: {{math|4<sup>2</sup> ≡ 16 ≡ 4 (mod 6)}}
: {{math|5<sup>2</sup> ≡ 25 ≡ 1 (mod 6)}}

From these computations, {{math|0}}, {{math|1}}, {{math|3}}, and {{math|4}} are idempotents of this ring, while {{math|2}} and {{math|5}} are not. This also demonstrates the decomposition properties described below: because {{math|3 + 4 ≡ 1 (mod 6)}}, there is a ring decomposition {{math|3'''Z''' / 6'''Z''' ⊕ 4'''Z''' / 6'''Z'''}}.  In {{math|3'''Z''' / 6'''Z'''}} the multiplicative identity is {{math|3 + 6'''Z'''}} and in {{math|4'''Z''' / 6'''Z'''}} the multiplicative identity is {{math|4 + 6'''Z'''}}.

=== Quotient of polynomial ring ===
Given a ring {{math|''R''}} and an element {{math|''f'' ∈ ''R''}} such that {{math|{{italics correction|''f''}}<sup>2</sup> ≠ 0}}, the [[quotient ring]]
: {{math|''R'' / ({{italics correction|''f''}}<sup>2</sup> − ''f'')}}
has the idempotent {{math|''f''}}. For example, this could be applied to {{math|''x'' ∈ '''Z'''[''x'']}}, or any [[polynomial]] {{math|''f'' ∈ ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>]}}.

=== Idempotents in the ring of split-quaternions ===
There is a [[circle]] of idempotents in the ring of [[split-quaternion]]s. Split quaternions have the structure of a [[algebra over a field|real algebra]], so elements can be written ''w'' + ''x''i + ''y''j + ''z''k over a [[basis (linear algebra)|basis]] {1, i, j, k}, with j<sup>2</sup> = k<sup>2</sup> = +1. For any &theta;, 
:<math>s = j \cos \theta + k \sin \theta</math> satisfies s<sup>2</sup> = +1 since j and k satisfy the [[anticommutative property]]. Now
:<math>(\frac{1+s}{2})^2 = \frac{1 + 2s + s^2}{4} = \frac{1+s}{2},</math> the idempotent property.

The element ''s'' is called a [[hyperbolic unit]] 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]] of [[split-quaternion#Hyperbolic units|hyperbolic units in split-quaternions]]. The same equality shows the idempotent property of <math>\frac{1 + s}{2}</math> where ''s'' is on the hyperboloid.

== Types of ring idempotents ==
A partial list of important types of idempotents includes:
* Two idempotents {{math|''a''}} and {{math|''b''}} are called '''orthogonal''' if {{math|1=''ab'' = ''ba'' = 0}}. If {{math|''a''}} is idempotent in the ring {{math|''R''}} (with [[Ring (mathematics)#Notes on the definition|unity]]), then so is {{math|1=''b'' = 1 − ''a''}}; moreover, {{math|''a''}} and {{math|''b''}} are orthogonal.
* An idempotent {{math|''a''}} in {{math|''R''}} is called a '''central idempotent''' if {{math|1=''ax'' = ''xa''}} for all {{math|''x''}} in {{math|''R''}}, that is, if {{math|''a''}} is in the [[center (ring theory)|center]] of {{math|''R''}}.
* A '''trivial idempotent''' refers to either of the elements {{math|0}} and {{math|1}}, which are always idempotent.
* A '''primitive idempotent''' of a ring {{math|''R''}} is a nonzero idempotent {{math|''a''}} such that {{math|''aR''}} is [[indecomposable module|indecomposable]] as a right {{math|''R''}}-module; that is, such that {{math|''aR''}} is not a [[direct sum of modules|direct sum]] of two [[zero module|nonzero]] [[submodule]]s. Equivalently, {{math|''a''}} is a primitive idempotent if it cannot be written as {{math|1=''a'' = ''e'' + ''f''}}, where {{math|''e''}} and {{math|''f''}} are nonzero orthogonal idempotents in {{math|''R''}}.
* A '''local idempotent''' is an idempotent {{math|''a''}} such that {{math|''aRa''}} is a [[local ring]].  This implies that {{math|''aR''}} is directly indecomposable, so local idempotents are also primitive.
* A '''right irreducible idempotent''' is an idempotent {{math|''a''}} for which {{math|''aR''}} is a [[simple module]].  By [[Schur's lemma]], {{math|1=End<sub>''R''</sub>(''aR'') = ''aRa''}} is a [[division ring]], and hence is a local ring, so right (and left) irreducible idempotents are local.
* A '''centrally primitive''' idempotent is a central idempotent {{math|''a''}} that cannot be written as the sum of two nonzero orthogonal central idempotents.
* An idempotent {{math|1=''a'' + ''I''}} in the quotient ring {{math|''R'' / ''I''}} is said to '''lift modulo {{math|''I''}}''' if there is an idempotent {{math|''b''}} in {{math|''R''}} such that {{math|1=''b'' + ''I'' = ''a'' + ''I''}}.
* An idempotent {{math|''a''}} of {{math|''R''}} is called a '''full idempotent''' if {{math|1=''RaR'' = ''R''}}.
* A '''separability idempotent'''; see ''[[Separable algebra]]''.

Any non-trivial idempotent {{math|''a''}} is a [[zero divisor]] (because {{math|1=''ab'' = 0}} with neither {{math|''a''}} nor {{math|''b''}} being zero, where {{math|1=''b'' = 1 − ''a''}}). This shows that [[integral domain]]s and [[division ring]]s do not have such idempotents. [[Local ring]]s also do not have such idempotents, but for a different reason. The only idempotent contained in the [[Jacobson radical]] of a ring is {{math|0}}.

== Rings characterized by idempotents ==
* A ring in which ''all'' elements are idempotent is called a [[Boolean ring]]. Some authors use the term "idempotent ring" for this type of ring. In such a ring, multiplication is [[commutative ring|commutative]] and every element is its own [[additive inverse]].
* A ring is [[semisimple ring|semisimple]] if and only if every right (or every left) [[ideal (ring theory)|ideal]] is generated by an idempotent.
* A ring is [[von Neumann regular ring|von Neumann regular]] if and only if every [[finitely generated module|finitely generated]] right (or every finitely generated left) ideal is generated by an idempotent.
* A ring for which the [[annihilator (ring theory)|annihilator]] {{math|''r''.Ann(''S'')}} every subset {{math|''S''}} of {{math|''R''}} is generated by an idempotent is called a [[Baer ring]]. If the condition only holds for all [[singleton (mathematics)|singleton]] subsets of {{math|''R''}}, then the ring is a right [[Rickart ring]]. Both of these types of rings are interesting even when they [[rng (algebra)|lack a multiplicative identity]].
* A ring in which all idempotents are [[Center (ring theory)|central]] is called an '''abelian ring'''.  Such rings need not be commutative.
* A ring is [[irreducible ring|directly irreducible]] if and only if {{math|0}} and {{math|1}} are the only central idempotents.
* A ring {{math|''R''}} can be written as {{math|1=''e''<sub>1</sub>''R'' ⊕ ''e''<sub>2</sub>''R'' ⊕ ... ⊕ ''e''<sub>''n''</sub>''R''}} with each {{math|''e''<sub>''i''</sub>}} a local idempotent if and only if {{math|''R''}} is a [[semiperfect ring]].
* A ring is called an '''[[SBI ring]]''' or '''Lift/rad''' ring if all idempotents of {{math|''R''}} lift modulo the [[Jacobson radical]].
* A ring satisfies the [[ascending chain condition]] on right direct summands if and only if the ring satisfies the [[descending chain condition]] on left direct summands if and only if every set of pairwise orthogonal idempotents is finite.
* If {{math|''a''}} is idempotent in the ring {{math|''R''}}, then {{math|''aRa''}} is again a ring, with multiplicative identity {{math|''a''}}.  The ring {{math|''aRa''}} is often referred to as a '''corner ring''' of {{math|''R''}}.  The corner ring arises naturally since the [[ring of endomorphisms]] {{math|1=End<sub>''R''</sub>(''aR'') ≅ ''aRa''}}.

== Role in decompositions ==
The idempotents of {{math|''R''}} have an important connection to decomposition of {{math|''R''}}-[[module (mathematics)|modules]].  If {{math|''M''}} is an {{math|''R''}}-module and {{math|1=''E'' = End<sub>''R''</sub>(''M'')}} is its [[ring of endomorphisms]], then {{math|1=''A'' ⊕ ''B'' = ''M''}} if and only if there is a unique idempotent {{math|''e''}} in {{math|''E''}} such that {{math|1=''A'' = ''eM''}} and {{math|1=''B'' = (1 − ''e'')''M''}}.  Clearly then, {{math|''M''}} is directly indecomposable if and only if {{math|0}} and {{math|1}} are the only idempotents in {{math|''E''}}.{{sfn|ps=none|Anderson|Fuller|1992|loc=p. 69–72}}

In the case when {{math|1=''M'' = ''R''}} (assumed unital), the endomorphism ring {{math|1=End<sub>''R''</sub>(''R'') = ''R''}}, where each [[endomorphism]] arises as left multiplication by a fixed ring element. With this modification of notation, {{math|1=''A'' ⊕ ''B'' = ''R''}} as right modules if and only if there exists a unique idempotent {{math|''e''}} such that {{math|1=''eR'' = ''A''}} and {{math|1=(1 − ''e'')''R'' = ''B''}}.  Thus every direct summand of {{math|''R''}} is generated by an idempotent.

If {{math|''a''}} is a central idempotent, then the corner ring {{math|1=''aRa'' = ''Ra''}} is a ring with multiplicative identity {{math|''a''}}. Just as idempotents determine the direct decompositions of {{math|''R''}} as a module, the central idempotents of {{math|''R''}} determine the decompositions of {{math|''R''}} as a [[Direct sum of modules|direct sum]] of rings. If {{math|''R''}} is the direct sum of the rings {{math|''R''<sub>1</sub>}}, ..., {{math|''R''<sub>''n''</sub>}}, then the identity elements of the rings {{math|''R''<sub>''i''</sub>}} are central idempotents in {{math|''R''}}, pairwise orthogonal, and their sum is {{math|1}}. Conversely, given central idempotents {{math|''a''<sub>1</sub>}}, ..., {{math|''a''<sub>''n''</sub>}} in {{math|''R''}} that are pairwise orthogonal and have sum {{math|1}}, then {{math|''R''}} is the direct sum of the rings {{math|''Ra''<sub>1</sub>}}, ..., {{math|''Ra''<sub>''n''</sub>}}. So in particular, every central idempotent {{math|''a''}} in {{math|''R''}} gives rise to a decomposition of {{math|''R''}} as a direct sum of the corner rings {{math|''aRa''}} and {{math|1=(1 − ''a'')''R''(1 − ''a'')}}.  As a result, a ring {{math|''R''}} is directly indecomposable as a ring if and only if the identity {{math|1}} is centrally primitive.

Working inductively, one can attempt to decompose {{math|1}} into a sum of centrally primitive elements.  If {{math|1}} 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 "{{math|''R''}}'' 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 ring|Noetherian]].  If a decomposition {{math|1=''R'' = ''c''<sub>1</sub>''R'' ⊕ ''c''<sub>2</sub>''R'' ⊕ ... ⊕ ''c''<sub>''n''</sub>''R''}} exists with each {{math|''c''<sub>''i''</sub>}} a centrally primitive idempotent, then {{math|''R''}} is a direct sum of the corner rings {{math|''c''<sub>''i''</sub>''Rc''<sub>''i''</sub>}}, each of which is ring irreducible.{{sfn|ps=none|Lam|2001|loc=p. 326}}

For [[associative algebra]]s or [[Jordan algebra]]s over a field, the [[Peirce decomposition]] is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements.

== Relation with involutions ==
If {{math|''a''}} is an idempotent of the endomorphism ring {{math|End<sub>''R''</sub>(''M'')}}, then the endomorphism {{math|1=''f'' = 1 − 2''a''}} is an {{math|''R''}}-module [[involution (mathematics)|involution]] of {{math|''M''}}. That is, {{math|''f''}} is an {{math|''R''}}-[[module homomorphism]] such that {{math|{{italics correction|''f''}}<sup>2</sup>}} is the identity endomorphism of {{math|''M''}}.

An idempotent element {{math|''a''}} of {{math|''R''}} and its associated involution {{math|''f''}} gives rise to two involutions of the module {{math|''R''}}, depending on viewing {{math|''R''}} as a left or right module. If {{math|''r''}} represents an arbitrary element of {{math|''R''}}, {{math|''f''}} can be viewed as a right {{math|''R''}}-module homomorphism {{math|1= ''r'' ↦ ''fr''}} so that {{math|1=''ffr'' = ''r''}}, or {{math|''f''}} can also be viewed as a left {{math|''R''}}-module homomorphism {{math|1=''r'' ↦ ''rf''}}, where {{math|1=''rff'' = ''r''}}.

This process can be reversed if {{math|2}} is an [[invertible element]] of {{math|''R''}}:{{efn|Rings in which {{math|2}} is not invertible are not difficult to find. The element {{math|2}} is not invertible in any ring of [[characteristic (algebra)|characteristic]] {{math|2}}, which includes [[Boolean ring]]s.{{clarify|reason=The restriction "invertible" should presumably be replaced by "cancellable": an example where {{math|2}} is not invertible, but this process works through division is in the ring of integers.|date=October 2023}}}} if {{math|''b''}} is an involution, then {{math|1=2<sup>−1</sup>(1 − ''b'')}} and {{math|1=2<sup>−1</sup>(1 + ''b'')}} are orthogonal idempotents, corresponding to {{math|''a''}} and {{math|1 − ''a''}}. Thus for a ring in which {{math|2}} is invertible, the idempotent elements [[bijection|correspond]] to involutions in a one-to-one manner.

== Category of ''R''-modules ==
Lifting idempotents also has major consequences for the [[category of modules|category of {{math|''R''}}-modules]]. All idempotents lift modulo {{math|''I''}} if and only if every {{math|''R''}} direct summand of {{math|1=''R''/''I''}} has a [[projective cover]] as an {{math|''R''}}-module.{{sfn|ps=none|Anderson|Fuller|1992|loc=p. 302}}  Idempotents always lift modulo [[nil ideal]]s and rings for which {{math|''R''}} is [[completion (ring theory)#Krull topology|{{math|''I''}}-adically complete]].

Lifting is most important when {{math|1=''I'' = J(''R'')}}, the [[Jacobson radical]] of {{math|''R''}}.  Yet another characterization of [[semiperfect ring]]s is that they are [[semilocal ring]]s whose idempotents lift modulo {{math|J(''R'')}}.{{sfn|ps=none|Lam|2001|loc=p. 336}}

== Lattice of idempotents ==
One may define a [[partial order]] on the idempotents of a ring as follows: if {{math|''a''}} and {{math|''b''}} are idempotents, we write {{math|1=''a'' ≤ ''b''}} if and only if {{math|1=''ab'' = ''ba'' = ''a''}}. With respect to this order, {{math|0}} is the smallest and {{math|1}} the largest idempotent. For orthogonal idempotents {{math|''a''}} and {{math|''b''}}, {{math|1=''a'' + ''b''}} is also idempotent, and we have {{math|1=''a'' ≤ ''a'' + ''b''}} and {{math|1=''b'' ≤ ''a'' + ''b''}}.  The [[atom (order theory)|atoms]] of this partial order are precisely the primitive idempotents.{{sfn|ps=none|Lam|2001|p=323}}

When the above partial order is restricted to the central idempotents of {{math|''R''}}, a [[lattice (order)|lattice]] structure, or even a [[Boolean algebra]] structure, can be given.  For two central idempotents {{math|''e''}} and {{math|''f''}}, the [[Boolean algebra#Operations|complement]] is given by
: {{math|1=¬''e'' = 1 − ''e''}},
the [[Meet (mathematics)|meet]] is given by
: {{math|1=''e'' ∧ ''f'' = ''ef''}}.
and the [[Join (mathematics)|join]] is given by
: {{math|1=''e'' ∨ ''f'' = ¬(¬''e'' ∧ ¬''f'') = ''e'' + ''f'' − ''ef''}}
The ordering now becomes simply {{math|1=''e'' ≤ ''f''}} if and only if {{math|1=''eR'' ⊆ {{italics correction|''f''}}''R''}}, and the join and meet satisfy {{math|1=(''e'' ∨ {{italics correction|''f''}})''R'' = ''eR'' + {{italics correction|''f''}}''R''}} and {{math|1=(''e'' ∧ {{italics correction|''f''}})''R'' = ''eR'' ∩ {{italics correction|''f''}}''R'' = (''eR'')({{italics correction|''f''}}''R'')}}.  It is shown in {{harvnb|Goodearl|1991|p=99}} that if {{math|''R''}} is [[von Neumann regular]] and right [[injective module#Self-injective rings|self-injective]], then the lattice is a [[complete lattice]].

== Notes ==
{{notelist}}

== Citations ==
{{reflist}}

== References ==
{{refbegin}}
* {{citation
   |last1=Anderson |first1=Frank Wylie
   |last2=Fuller |first2=Kent R
   |title=Rings and Categories of Modules
   |publisher=[[Springer-Verlag]] |location=Berlin, New York
   |isbn=978-0-387-97845-1
   |year=1992
  }}
* [http://foldoc.org/idempotent idempotent] at [[FOLDOC]]
* {{citation
   |last1=Goodearl |first1=K. R.
   |title=von Neumann regular rings
   |edition=2
   |publisher=Robert E. Krieger Publishing Co. Inc.
   |place=Malabar, FL
   |year=1991
   |pages=xviii+412
   |isbn=0-89464-632-X
   |mr=1150975
  }}
* {{citation
   |last1=Hazewinkel |first1=Michiel
   |last2=Gubareni |first2=Nadiya
   |last3=Kirichenko |first3=V. V.
   |title=Algebras, rings and modules. Vol. 1
   |series=Mathematics and its Applications
   |volume=575
   |publisher=Kluwer Academic Publishers
   |place=Dordrecht
   |year=2004
   |pages=xii+380
   |isbn=1-4020-2690-0
   |mr=2106764
  }}
* {{citation
   |last1=Lam |first1=T. Y.
   |title=A first course in noncommutative rings
   |series=Graduate Texts in Mathematics
   |volume=131
   |edition=2
   |publisher=Springer-Verlag
   |place=New York
   |year=2001
   |pages=xx+385
   |isbn=0-387-95183-0
   |mr=1838439
   |doi=10.1007/978-1-4419-8616-0
  }}
* {{Lang Algebra |edition=3 |page=443}}
* {{citation
   |last1=Peirce |first1=Benjamin
   |url=http://www.math.harvard.edu/history/peirce_algebra/index.html
   |title=Linear Associative Algebra
   |year=1870
  }}
* {{citation
   |last1=Polcino Milies |first1=César
   |last2=Sehgal |first2=Sudarshan K.
   |title=An introduction to group rings
   |series=Algebras and Applications
   |volume=1
   |publisher=Kluwer Academic Publishers
   |place=Dordrecht
   |year=2002
   |pages=xii+371
   |isbn=1-4020-0238-6
   |mr=1896125
   |doi=10.1007/978-94-010-0405-3
  }}
{{refend}}

[[Category:Ring theory]]