Idempotent (ring theory)
In ring theory (part of abstract algebra) an idempotent element, or simply an idempotent, of a ring is an element a such that a2 = a. That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that a = a2 = a3 = a4 = ... = an for any positive integer 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 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 mod n, where n is squarefree. By the Chinese remainder theorem, this ring factors into the direct product of rings of integers mod p. Now each of these factors is a field, so it is clear that the factor's only idempotents will be 0 and 1. That is, each factor has two idempotents. So if there are m factors, there will be 2m idempotents.
We can check this for the integers mod 6, R = Z/6Z. Since 6 has two factors (2 and 3) it should have 22 idempotents.
From these computations, 0, 1, 3, and 4 are idempotents of this ring, while 2 and 5 are not. This also demonstrates the decomposition properties described below: because 3 + 4 = 1 (mod 6), there is a ring decomposition 3Z/6Z ⊕ 4Z/6Z. In 3Z/6Z the identity is 3+6Z and in 4Z/6Z the identity is 4+6Z.
Any non-trivial idempotent a is a zero divisor (because ab = 0 with neither a nor b being zero, where b = 1 − a). 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 0.
The idempotents of R have an important connection to decomposition of R modules. If M is an R module and E = EndR(M) is its ring of endomorphisms, then A ⊕ B = M if and only if there is a unique idempotent e in E such that A = e(M) and B = (1 − e) (M). Clearly then, M is directly indecomposable if and only if 0 and 1 are the only idempotents in E.
In the case when M = R the endomorphism ring EndR(R) = R, where each endomorphism arises as left multiplication by a fixed ring element. With this modification of notation, A ⊕ B = R as right modules if and only if there exists a unique idempotent e such that eR = A and (1 − e)R = B. Thus every direct summand of R is generated by an idempotent.
If a is a central idempotent, then the corner ring aRa = Ra is a ring with multiplicative identity a. Just as idempotents determine the direct decompositions of R as a module, the central idempotents of R determine the decompositions of R as a direct sum of rings. If R is the direct sum of the rings R1,...,Rn, then the identity elements of the rings Ri are central idempotents in R, pairwise orthogonal, and their sum is 1. Conversely, given central idempotents a1,...,an in R that are pairwise orthogonal and have sum 1, then R is the direct sum of the rings Ra1,…,Ran. So in particular, every central idempotent a in R gives rise to a decomposition of R as a direct sum of the corner rings aRa and (1 − a)R(1 − a). As a result, a ring R is directly indecomposable as a ring if and only if the identity 1 is centrally primitive.
Working inductively, one can attempt to decompose 1 into a sum of centrally primitive elements. If 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 "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. If a decomposition R = c1R ⊕ c2R ⊕ ... ⊕ cnR exists with each ci a centrally primitive idempotent, then R is a direct sum of the corner rings ciRci, each of which is ring irreducible.
If a is an idempotent of the endomorphism ring EndR(M), then the endomorphism f = 1 − 2a is an R module involution of M. That is, f is an R homomorphism such that f 2 is the identity endomorphism of M.
An idempotent element a of R and its associated involution f gives rise to two involutions of the module R, depending on viewing R as a left or right module. If r represents an arbitrary element of R, f can be viewed as a right R-homomorphism r ↦ fr so that ffr = r, or f can also be viewed as a left R module homomorphism r ↦ rf, where rff = r.
This process can be reversed if 2 is an invertible element of R: if b is an involution, then 2−1(1 − b) and 2−1(1 + b) are orthogonal idempotents, corresponding to a and 1 − a. Thus for a ring in which 2 is invertible, the idempotent elements correspond to involutions in a one-to-one manner.
Lifting idempotents also has major consequences for the category of R modules. All idempotents lift modulo I if and only if every R direct summand of R/I has a projective cover as an R module. Idempotents always lift modulo nil ideals and rings for which R/I is I-adically complete.
One may define a partial order on the idempotents of a ring as follows: if a and b are idempotents, we write a ≤ b if and only if ab = ba = a. With respect to this order, 0 is the smallest and 1 the largest idempotent. For orthogonal idempotents a and b, a + b is also idempotent, and we have a ≤ a + b and b ≤ a + b. The atoms of this partial order are precisely the primitive idempotents. (Lam 2001, p. 323)
When the above partial order is restricted to the central idempotents of R, a lattice structure, or even a Boolean algebra structure, can be given. For two central idempotents e and f the complement ¬e = 1 − e and the join and meet are given by
The ordering now becomes simply e ≤ f if and only if eR ⊆ fR, and the join and meet satisfy (e ∨ f)R = eR + fR and (e ∧ f)R = eR ∩ fR = (eR)(fR). It is shown in (Goodearl 1991, p. 99) that if R is von Neumann regular and right self-injective, then the lattice is a complete lattice.