Idempotent (ring theory)
Template:Short description In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element Template:Math such that Template:Math.Template:SfnTemplate:Efn That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that Template:Math for any positive integer Template:Math. 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 modules, and connected to 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.
ExamplesEdit
Quotients of ZEdit
One may consider the [[ring of integers modulo n|ring of integers modulo Template:Math]], where Template:Math is square-free. By the Chinese remainder theorem, this ring factors into the product of rings of integers modulo Template:Math, where Template:Math is prime. Now each of these factors is a field, so it is clear that the factors' only idempotents will be Template:Math and Template:Math. That is, each factor has two idempotents. So if there are Template:Math factors, there will be Template:Math idempotents.
We can check this for the integers Template:Math, Template:Math. Since Template:Math has two prime factors (Template:Math and Template:Math) it should have Template:Math idempotents.
From these computations, Template:Math, Template:Math, Template:Math, and Template:Math are idempotents of this ring, while Template:Math and Template:Math are not. This also demonstrates the decomposition properties described below: because Template:Math, there is a ring decomposition Template:Math. In Template:Math the multiplicative identity is Template:Math and in Template:Math the multiplicative identity is Template:Math.
Quotient of polynomial ringEdit
Given a ring Template:Math and an element Template:Math such that Template:Math, the quotient ring
has the idempotent Template:Math. For example, this could be applied to Template:Math, or any polynomial Template:Math.
Idempotents in the ring of split-quaternionsEdit
There is a circle of idempotents in the ring of split-quaternions. Split quaternions have the structure of a real algebra, so elements can be written w + xi + yj + zk over a basis {1, i, j, k}, with j2 = k2 = +1. For any θ,
- <math>s = j \cos \theta + k \sin \theta</math> satisfies s2 = +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 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 idempotentsEdit
A partial list of important types of idempotents includes:
- Two idempotents Template:Math and Template:Math are called orthogonal if Template:Math. If Template:Math is idempotent in the ring Template:Math (with unity), then so is Template:Math; moreover, Template:Math and Template:Math are orthogonal.
- An idempotent Template:Math in Template:Math is called a central idempotent if Template:Math for all Template:Math in Template:Math, that is, if Template:Math is in the center of Template:Math.
- A trivial idempotent refers to either of the elements Template:Math and Template:Math, which are always idempotent.
- A primitive idempotent of a ring Template:Math is a nonzero idempotent Template:Math such that Template:Math is indecomposable as a right Template:Math-module; that is, such that Template:Math is not a direct sum of two nonzero submodules. Equivalently, Template:Math is a primitive idempotent if it cannot be written as Template:Math, where Template:Math and Template:Math are nonzero orthogonal idempotents in Template:Math.
- A local idempotent is an idempotent Template:Math such that Template:Math is a local ring. This implies that Template:Math is directly indecomposable, so local idempotents are also primitive.
- A right irreducible idempotent is an idempotent Template:Math for which Template:Math is a simple module. By Schur's lemma, Template:Math 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 Template:Math that cannot be written as the sum of two nonzero orthogonal central idempotents.
- An idempotent Template:Math in the quotient ring Template:Math is said to lift modulo Template:Math if there is an idempotent Template:Math in Template:Math such that Template:Math.
- An idempotent Template:Math of Template:Math is called a full idempotent if Template:Math.
- A separability idempotent; see Separable algebra.
Any non-trivial idempotent Template:Math is a zero divisor (because Template:Math with neither Template:Math nor Template:Math being zero, where Template:Math). This shows that integral domains and division rings do not have such idempotents. Local rings also do not have such idempotents, but for a different reason. The only idempotent contained in the Jacobson radical of a ring is Template:Math.
Rings characterized by idempotentsEdit
- 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 and every element is its own additive inverse.
- A ring is semisimple if and only if every right (or every left) ideal is generated by an idempotent.
- A ring is von Neumann regular 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 Template:Math every subset Template:Math of Template:Math is generated by an idempotent is called a Baer ring. If the condition only holds for all singleton subsets of Template:Math, 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 Template:Math and Template:Math are the only central idempotents.
- A ring Template:Math can be written as Template:Math with each Template:Math a local idempotent if and only if Template:Math is a semiperfect ring.
- A ring is called an SBI ring or Lift/rad ring if all idempotents of Template:Math 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 Template:Math is idempotent in the ring Template:Math, then Template:Math is again a ring, with multiplicative identity Template:Math. The ring Template:Math is often referred to as a corner ring of Template:Math. The corner ring arises naturally since the ring of endomorphisms Template:Math.
Role in decompositionsEdit
The idempotents of Template:Math have an important connection to decomposition of Template:Math-modules. If Template:Math is an Template:Math-module and Template:Math is its ring of endomorphisms, then Template:Math if and only if there is a unique idempotent Template:Math in Template:Math such that Template:Math and Template:Math. Clearly then, Template:Math is directly indecomposable if and only if Template:Math and Template:Math are the only idempotents in Template:Math.Template:Sfn
In the case when Template:Math (assumed unital), the endomorphism ring Template:Math, where each endomorphism arises as left multiplication by a fixed ring element. With this modification of notation, Template:Math as right modules if and only if there exists a unique idempotent Template:Math such that Template:Math and Template:Math. Thus every direct summand of Template:Math is generated by an idempotent.
If Template:Math is a central idempotent, then the corner ring Template:Math is a ring with multiplicative identity Template:Math. Just as idempotents determine the direct decompositions of Template:Math as a module, the central idempotents of Template:Math determine the decompositions of Template:Math as a direct sum of rings. If Template:Math is the direct sum of the rings Template:Math, ..., Template:Math, then the identity elements of the rings Template:Math are central idempotents in Template:Math, pairwise orthogonal, and their sum is Template:Math. Conversely, given central idempotents Template:Math, ..., Template:Math in Template:Math that are pairwise orthogonal and have sum Template:Math, then Template:Math is the direct sum of the rings Template:Math, ..., Template:Math. So in particular, every central idempotent Template:Math in Template:Math gives rise to a decomposition of Template:Math as a direct sum of the corner rings Template:Math and Template:Math. As a result, a ring Template:Math is directly indecomposable as a ring if and only if the identity Template:Math is centrally primitive.
Working inductively, one can attempt to decompose Template:Math into a sum of centrally primitive elements. If Template:Math 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 "Template:Math 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. If a decomposition Template:Math exists with each Template:Math a centrally primitive idempotent, then Template:Math is a direct sum of the corner rings Template:Math, each of which is ring irreducible.Template:Sfn
For associative algebras or Jordan algebras over a field, the Peirce decomposition is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements.
Relation with involutionsEdit
If Template:Math is an idempotent of the endomorphism ring Template:Math, then the endomorphism Template:Math is an Template:Math-module involution of Template:Math. That is, Template:Math is an Template:Math-module homomorphism such that Template:Math is the identity endomorphism of Template:Math.
An idempotent element Template:Math of Template:Math and its associated involution Template:Math gives rise to two involutions of the module Template:Math, depending on viewing Template:Math as a left or right module. If Template:Math represents an arbitrary element of Template:Math, Template:Math can be viewed as a right Template:Math-module homomorphism Template:Math so that Template:Math, or Template:Math can also be viewed as a left Template:Math-module homomorphism Template:Math, where Template:Math.
This process can be reversed if Template:Math is an invertible element of Template:Math:Template:Efn if Template:Math is an involution, then Template:Math and Template:Math are orthogonal idempotents, corresponding to Template:Math and Template:Math. Thus for a ring in which Template:Math is invertible, the idempotent elements correspond to involutions in a one-to-one manner.
Category of R-modulesEdit
Lifting idempotents also has major consequences for the [[category of modules|category of Template:Math-modules]]. All idempotents lift modulo Template:Math if and only if every Template:Math direct summand of Template:Math has a projective cover as an Template:Math-module.Template:Sfn Idempotents always lift modulo nil ideals and rings for which Template:Math is [[completion (ring theory)#Krull topology|Template:Math-adically complete]].
Lifting is most important when Template:Math, the Jacobson radical of Template:Math. Yet another characterization of semiperfect rings is that they are semilocal rings whose idempotents lift modulo Template:Math.Template:Sfn
Lattice of idempotentsEdit
One may define a partial order on the idempotents of a ring as follows: if Template:Math and Template:Math are idempotents, we write Template:Math if and only if Template:Math. With respect to this order, Template:Math is the smallest and Template:Math the largest idempotent. For orthogonal idempotents Template:Math and Template:Math, Template:Math is also idempotent, and we have Template:Math and Template:Math. The atoms of this partial order are precisely the primitive idempotents.Template:Sfn
When the above partial order is restricted to the central idempotents of Template:Math, a lattice structure, or even a Boolean algebra structure, can be given. For two central idempotents Template:Math and Template:Math, the complement is given by
the meet is given by
and the join is given by
The ordering now becomes simply Template:Math if and only if Template:Math, and the join and meet satisfy Template:Math and Template:Math. It is shown in Template:Harvnb that if Template:Math is von Neumann regular and right self-injective, then the lattice is a complete lattice.