Idempotent (ring theory)

In abstract algebra, more specifically in ring theory, an idempotent element, or simply an idempotent, of a ring is an element a such that a² = a.^[1] 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 tied with decompositions of modules, as well as 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.

Examples

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 only idempotents will be 0 and 1. That is, each factor has 2 idempotents. So if there are m factors, there will be 2^m idempotents.

We can check this for the integers mod 6, R = Z/6Z. Since 6 has 2 factors (2 and 3) it should have 2² idempotents.

0² = 0 = 0 (mod 6)

1² = 1 = 1 (mod 6)

2² = 4 = 4 (mod 6)

3² = 9 = 3 (mod 6)

4² = 16 = 4 (mod 6)

5² = 25 = 1 (mod 6)

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.

Other examples

There is a catenoid of idempotents in the coquaternion ring.

Types of ring idempotents

A partial list of important types of idempotents includes:

• Two idempotents a and b are called orthogonal if ab = ba = 0. If a is idempotent in the ring R (with unity), then so is b = 1 − a; moreover, a and b are orthogonal.
• An idempotent a in R is called a central idempotent if ax = xa for all x in R.
• A trivial idempotent refers to either of the elements 0 and 1, which are always idempotent.
• A primitive idempotent is an idempotent a such that aR is directly indecomposable.
• A local idempotent is an idempotent a such that aRa is a local ring. This implies that aR is directly indecomposable, so local idempotents are also primitive.
• A right irreducible idempotent is an idempotent a for which aR is a simple module. By Schur's lemma, End_R(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 a that cannot be written as the sum of two nonzero orthogonal central idempotents.
• An idempotent e + I in the quotient ring R/I is said to lift modulo I if there is an idempotent f in R such that f + I = e + I.
• An idempotent of e of R is called a full idempotent if ReR=R.
• A separability idempotent; see separable algebra.

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 don't have such idempotents. Local rings also don't have such idempotents, but for a different reason. The only idempotent contained in the Jacobson radical of a ring is 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 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 every subset S of R is generated by an idempotent is called a Baer ring. If the condition only holds for all singleton subsets of R, 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 0 and 1 are the only central idempotents.
• A ring R can be written as e₁R ⊕ e₂R ⊕ ... ⊕ e_nR with each e_i a local idempotent if and only if R is a semiperfect ring.
• A ring is called an SBI ring or Lift/rad ring if all idempotents of 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 a is idempotent in the ring R, then aRa is again a ring, with multiplicative identity a. The ring aRa is often referred to as a corner ring of R. The corner ring arises naturally since the ring of endomorphisms End_R(aR) ≅ aRa.

Role in decompositions

The idempotents of R have an important connection to decomposition of R modules. If M is an R module and E = End_R(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.^[2]

In the case when M = R the endomorphism ring End_R(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 module 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,...,R_n, then the identity elements of the rings R_i are central idempotents in R, pairwise orthogonal, and their sum is 1. Conversely, given central idempotents a₁,...,a_n in R that are pairwise orthogonal and have sum 1, then R is the direct sum of the rings Ra₁,…,Ra_n. 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 ⊕ ... ⊕ c_nR exists with each c_i a centrally primitive idempotent, then R is a direct sum of the corner rings c_iRc_i, each of which is ring irreducible.^[3]

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 involutions

If a is an idempotent of the endomorphism ring End_R(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:^[4] 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.

Category of R modules

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.^[5] Idempotents always lift modulo nil ideals and rings for which R/I is I-adically complete.

Lifting is most important when I = J(R), the Jacobson radical of R. Yet another characterization of semiperfect rings is that they are semilocal rings whose idempotents lift modulo J(R).^[6]

Lattice of ideals

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 can be given. For two central idempotents e and f the complement ¬e = 1 − e and the join and meet are given by

e ∨ f = e + f − ef

and

e ∧ f = ef.

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.

Notes

1. ↑ See Hazewinkel et al. (2004), p. 2.
2. ↑ Anderson & Fuller 1992, p.69-72.
3. ↑ Lam 2001, p.326.
4. ↑ Rings in which 2 is not invertible are not hard to find. The element 2 is not invertible in any Boolean algebra, nor in any ring of characteristic 2.
5. ↑ Anderson & Fuller 1992, p.302.
6. ↑ Lam 2001, p.336.

References