Inring theory, a branch ofmathematics, anidempotent element or simplyidempotent of aring is an elementa such thata² =a.^[1][a] That is, the element isidempotent under the ring's multiplication.Inductively then, one can also conclude thata =a² =a³ =a⁴ = ... =aⁿ for any positiveintegern. For example, an idempotent element of amatrix ring is precisely anidempotent matrix.

For general rings, elements idempotent under multiplication are involved in decompositions ofmodules, and connected tohomological properties of the ring. InBoolean algebra, the main objects of study are rings in which all elements are idempotent under both addition and multiplication.

One may consider thering of integers modulon, wheren issquare-free. By theChinese remainder theorem, this ring factors into theproduct of rings of integers modulo p, wherep isprime. Now each of these factors is afield, so it is clear that the factors' only idempotents will be0 and1. That is, each factor has two idempotents. So if there arem factors, there will be2^m idempotents.

We can check this for the integersmod 6,R =Z / 6Z. Since6 has two prime factors (2 and3) it should have2² 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, and4 are idempotents of this ring, while2 and5 are not. This also demonstrates the decomposition properties described below: because3 + 4 ≡ 1 (mod 6), there is a ring decomposition3Z / 6Z ⊕ 4Z / 6Z. In3Z / 6Z the multiplicative identity is3 + 6Z and in4Z / 6Z the multiplicative identity is4 + 6Z.

Given a ringR and an elementf ∈R such thatf² ≠ 0, thequotient ring

R / (f² −f)

has the idempotentf. For example, this could be applied tox ∈Z[x], or anypolynomialf ∈k[x₁, ...,x_n].

There is acircle of idempotents in the ring ofsplit-quaternions. Split quaternions have the structure of areal algebra, so elements can be writtenw +xi +yj +zk over abasis {1, i, j, k}, with j² = k² = +1. For any θ,

${\displaystyle s=j\cos \theta +k\sin \theta }$ satisfies s² = +1 since j and k satisfy theanticommutative property. Now

${\displaystyle {\left(\frac{1+s}{2}\right)}^2=\frac{1+2s+s^2}{4}=\frac{1+s}{2},}$ the idempotent property.

The elements is called ahyperbolic unit and so far, the i-coordinate has been taken as zero. When this coordinate is non-zero, then there is ahyperboloid of one sheet ofhyperbolic units in split-quaternions. The same equality shows the idempotent property of${\displaystyle \frac{1+s}{2}}$ wheres is on the hyperboloid.

A partial list of important types of idempotents includes:

• Two idempotentsa andb are calledorthogonal ifab =ba = 0. Ifa is idempotent in the ringR (withunity), then so isb = 1 −a; moreover,a andb are orthogonal.
• An idempotenta inR is called acentral idempotent ifax =xa for allx inR, that is, ifa is in thecenter ofR.
• Atrivial idempotent refers to either of the elements0 and1, which are always idempotent.
• Aprimitive idempotent of a ringR is a nonzero idempotenta such thataR isindecomposable as a rightR-module; that is, such thataR is not adirect sum of twononzerosubmodules. Equivalently,a is a primitive idempotent if it cannot be written asa =e +f, wheree andf are nonzero orthogonal idempotents inR.
• Alocal idempotent is an idempotenta such thataRa is alocal ring. This implies thataR is directly indecomposable, so local idempotents are also primitive.
• Aright irreducible idempotent is an idempotenta for whichaR is asimple module. BySchur's lemma,End_R(aR) =aRa is adivision ring, and hence is a local ring, so right (and left) irreducible idempotents are local.
• Acentrally primitive idempotent is a central idempotenta that cannot be written as the sum of two nonzero orthogonal central idempotents.
• An idempotenta +I in the quotient ringR /I is said tolift moduloI if there is an idempotentb inR such thatb +I =a +I.
• An idempotenta ofR is called afull idempotent ifRaR =R.
• Aseparability idempotent; seeSeparable algebra.

Any non-trivial idempotenta is azero divisor (becauseab = 0 with neithera norb being zero, whereb = 1 −a). This shows thatintegral domains anddivision rings do not have such idempotents.Local rings also do not have such idempotents, but for a different reason. The only idempotent contained in theJacobson radical of a ring is0.

• A ring in whichall elements are idempotent is called aBoolean ring. Some authors use the term "idempotent ring" for this type of ring. In such a ring, multiplication iscommutative and every element is its ownadditive inverse.
• A ring issemisimple if and only if every right (or every left)ideal is generated by an idempotent.
• A ring isvon Neumann regular if and only if everyfinitely generated right (or every finitely generated left) ideal is generated by an idempotent.
• A ring for which theannihilatorr.Ann(S) every subsetS ofR is generated by an idempotent is called aBaer ring. If the condition only holds for allsingleton subsets ofR, then the ring is a rightRickart ring. Both of these types of rings are interesting even when theylack a multiplicative identity.
• A ring in which all idempotents arecentral is called anabelian ring. Such rings need not be commutative.
• A ring isdirectly irreducible if and only if0 and1 are the only central idempotents.
• A ringR can be written ase₁R ⊕e₂R ⊕ ... ⊕e_nR with eache_i a local idempotent if and only ifR is asemiperfect ring.
• A ring is called anSBI ring orLift/rad ring if all idempotents ofR lift modulo theJacobson radical.
• A ring satisfies theascending chain condition on right direct summands if and only if the ring satisfies thedescending chain condition on left direct summands if and only if every set of pairwise orthogonal idempotents is finite.
• Ifa is idempotent in the ringR, thenaRa is again a ring, with multiplicative identitya. The ringaRa is often referred to as acorner ring ofR. The corner ring arises naturally since thering of endomorphismsEnd_R(aR) ≅aRa.

The idempotents ofR have an important connection to decomposition ofR-modules. IfM is anR-module andE = End_R(M) is itsring of endomorphisms, thenA ⊕B =M if and only if there is a unique idempotente inE such thatA =eM andB = (1 −e)M. Clearly then,M is directly indecomposable if and only if0 and1 are the only idempotents inE.^[2]

In the case whenM =R (assumed unital), the endomorphism ringEnd_R(R) =R, where eachendomorphism 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 idempotente such thateR =A and(1 −e)R =B. Thus every direct summand ofR is generated by an idempotent.

Ifa is a central idempotent, then the corner ringaRa =Ra is a ring with multiplicative identitya. Just as idempotents determine the direct decompositions ofR as a module, the central idempotents ofR determine the decompositions ofR as adirect sum of rings. IfR is the direct sum of the ringsR₁, ...,R_n, then the identity elements of the ringsR_i are central idempotents inR, pairwise orthogonal, and their sum is1. Conversely, given central idempotentsa₁, ...,a_n inR that are pairwise orthogonal and have sum1, thenR is the direct sum of the ringsRa₁, ...,Ra_n. So in particular, every central idempotenta inR gives rise to a decomposition ofR as a direct sum of the corner ringsaRa and(1 −a)R(1 −a). As a result, a ringR is directly indecomposable as a ring if and only if the identity1 is centrally primitive.

Working inductively, one can attempt to decompose1 into a sum of centrally primitive elements. If1 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 rightNoetherian. If a decompositionR =c₁R ⊕c₂R ⊕ ... ⊕c_nR exists with eachc_i a centrally primitive idempotent, thenR is a direct sum of the corner ringsc_iRc_i, each of which is ring irreducible.^[3]

Forassociative algebras orJordan algebras over a field, thePeirce decomposition is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements.

Ifa is an idempotent of a ringR, thenf = 1 − 2a equals its square. So, for every leftR-module, the multiplication byf is aninvolution ofM; that is, it is anR-module homomorphism such thatf² is the identity endomorphism ofM.

If⁠${\displaystyle M}$⁠ is an⁠${\displaystyle R}$⁠-bimodule, and, in particular, if⁠${\displaystyle M=R}$⁠, the left and the right multiplications with⁠${\displaystyle f}$⁠ gives rise to two involutions of the module.

Conversely, ifb is an element of⁠${\displaystyle R}$⁠ such that⁠${\displaystyle b^2=1}$⁠, then⁠${\displaystyle (1-b{)}^2=2(1-b)}$⁠, and, if2 is aninvertible element inR,a=2⁻¹(1 −b) is an idempotent such thatb = 1 − 2a. Thus, for a ring in which2 is invertible, the idempotent elements are inone-to-one correspondence with the elements whose square is 1.

Lifting idempotents also has major consequences for thecategory ofR-modules. All idempotents lift moduloI if and only if everyR direct summand ofR/I has aprojective cover as anR-module.^[4] Idempotents always lift modulonil ideals and rings for whichR isI-adically complete.

Lifting is most important whenI = J(R), theJacobson radical ofR. Yet another characterization ofsemiperfect rings is that they aresemilocal rings whose idempotents lift moduloJ(R).^[5]

One may define apartial order on the idempotents of a ring as follows: ifa andb are idempotents, we writea ≤b if and only ifab =ba =a. With respect to this order,0 is the smallest and1 the largest idempotent. For orthogonal idempotentsa andb,a +b is also idempotent, and we havea ≤a +b andb ≤a +b. Theatoms of this partial order are precisely the primitive idempotents.^[6]

When the above partial order is restricted to the central idempotents ofR, alattice structure, or even aBoolean algebra structure, can be given. For two central idempotentse andf, thecomplement is given by

¬e = 1 −e,

themeet is given by

e ∧f =ef.

and thejoin is given by

e ∨f = ¬(¬e ∧ ¬f) =e +f −ef

The ordering now becomes simplye ≤f if and only ifeR ⊆fR, and the join and meet satisfy(e ∨f)R =eR +fR and(e ∧f)R =eR ∩fR = (eR)(fR). It is shown inGoodearl 1991, p. 99 that ifR isvon Neumann regular and rightself-injective, then the lattice is acomplete lattice.

• Ring theory

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