In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element such that . That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that for any positive integer . 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.
One may consider the ring of integers modulo, where is squarefree. By the Chinese remainder theorem, this ring factors into the product of rings of integers modulo , where is prime. Now each of these factors is a field, 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 .
Given a ring and an element such that, the quotient ring
has the idempotent . For example, this could be applied to, or any polynomial .
There is a hyperboloid of idempotents in the split-quaternion ring.
A partial list of important types of idempotents includes:
Any non-trivial idempotent is a zero divisor (because with neither nor being zero, where). 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 .
The idempotents of have an important connection to decomposition of -modules. If is an -module and is its ring of endomorphisms, 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 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 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. 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 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.
If is an idempotent of the endomorphism ring, then the endomorphism is an -module involution of . That is, is an -module homomorphism 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 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.
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 as an -module. Idempotents always lift modulo nil ideals and rings for which is -adically complete.
Lifting is most important when, the Jacobson radical of . Yet another characterization of semiperfect rings is that they are semilocal rings whose idempotents lift modulo .
One may define a partial order 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 of this partial order are precisely the primitive idempotents.
When the above partial order is restricted to the central idempotents of, a lattice structure, or even a Boolean algebra structure, can be given. For two central idempotents and, the complement is given by
,the meet is given by
.and the join 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 and right self-injective, then the lattice is a complete lattice.