An element x of a ring is called an idempotent element, or simply an idempotentMathworldPlanetmath if x2=x.

The set of idempotents of a ring can be partially ordered by putting ef iff e=ef=fe.

The element 0 is a minimum element in this partial orderMathworldPlanetmath. If the ring has an identity elementMathworldPlanetmath, 1, then 1 is a maximum element in this partial order.

Since the above definitions refer only to the multiplicative structure of the ring, they also hold for semigroupsPlanetmathPlanetmath (with the proviso, of course, that a semigroup may have neither a zero elementMathworldPlanetmath nor an identity element). In the special case of a semilattice, this partial order is the same as the one described in the entry for semilattice.

If a ring has an identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, then 1-e is always an idempotent whenever e is an idempotent, and e(1-e)=(1-e)e=0.

In a ring with an identity, two idempotents e and f are called a pair of orthogonal idempotents if e+f=1, and ef=fe=0. Obviously, this is just a fancy way of saying that f=1-e.

More generally, a set {e1,e2,,en} of idempotents is called a complete set of orthogonal idempotents if eiej=ejei=0 whenever ij and if 1=e1+e2++en.

If {e1,e2,,en} is a complete set of orthogonal idempotents, and in addition each ei is in the centre of R, then each Rei is a subring, and


Conversely, whenever R1×R2××Rn is a direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of rings with identities, write ei for the element of the product corresponding to the identity element of Ri. Then {e1,e2,,en} is a complete set of central orthogonal idempotents of the product ring.

When a complete set of orthogonal idempotents is not central, there is a more complicated : see the entry on the Peirce decomposition for the details.

Title idempotent
Canonical name Idempotent
Date of creation 2013-03-22 13:07:27
Last modified on 2013-03-22 13:07:27
Owner mclase (549)
Last modified by mclase (549)
Numerical id 11
Author mclase (549)
Entry type Definition
Classification msc 16U99
Classification msc 20M99
Synonym idempotent element
Related topic Semilattice
Related topic Idempotency
Defines orthogonal idempotents
Defines complete set of orthogonal idempotents