central idempotent

Let R be a ring. An element eR is called a central idempotent if it is an idempotentMathworldPlanetmathPlanetmath and is in the center Z(R) of R.

It is well-known that if eR is an idempotent, then eRe has the structureMathworldPlanetmath of a ring with unity, with e being the unity. Thus, if e is central, eRe=eR=Re is a ring with unity e.

It is easy to see that the operationMathworldPlanetmath of ring multiplication preserves central idempotency: if e,f are central idempotents, so is ef. In addition, if R has a multiplicative identity 1, then f:=1-e is also a central idempotent. Furthermore, we may characterize central idempotency in a ring with 1 as follows:

Proposition 1.

An idempotent e in a ring R with 1 is central iff eRf=fRe=0, where f=1-e.


If e is central, then clearly eRf=fRe=0. Conversely, for any rR, we have er=er-erf=er(1-f)=ere=(1-f)re=re-fre=re. ∎

Another interesting fact about central idempotents in a ring with unity is the following:

Proposition 2.

The set C of all central idempotents of a ring R with 1 has the structure of a Boolean ringMathworldPlanetmath.


First, note that 0,1C. Next, for e,fC, we define addition and multiplication on C as follows:

ef:=e+f-ef  and  ef:=ef.

As discussed above, and are well-defined (as C is closed under these operations). In addition, for any e,f,gC, we have

  1. 1.

    (C,1,) is a commutative monoid, in which every element is an idempotent (with respect to ). This fact is clear.

  2. 2.

    is commutativePlanetmathPlanetmath, since CZ(R).

  3. 3.

    is associative:

    e(fg) = e+(f+g-fg)-e(f+g-fg)
    = e+f+g-ef-fg-eg+efg
    = (e+f-ef)+g-(e+f-ef)g
    = (ef)g.
  4. 4.

    distributes over : we only need to show left distributivity (since is commutative by 1 above):

    e(fg) = e(f+g-fg)=ef+eg-efg
    = ef+eg-eefg=ef+eg-efeg
    = efeg=(ef)(eg).

This shows that (C,0,1,,) is a Boolean ring. ∎

Title central idempotent
Canonical name CentralIdempotent
Date of creation 2013-03-22 19:13:07
Last modified on 2013-03-22 19:13:07
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 16U99
Classification msc 20M99