central idempotent (original) (raw)

It is well-known that if e∈R is an idempotent, then e⁢R⁢e has the structure of a ring with unity, with e being the unity. Thus, if e is central, e⁢R⁢e=e⁢R=R⁢e is a ring with unity e.

It is easy to see that the operation of ring multiplication preserves central idempotency: if e,f are central idempotents, so is e⁢f. 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 e⁢R⁢f=f⁢R⁢e=0, where f=1-e.

Proof.

If e is central, then clearly e⁢R⁢f=f⁢R⁢e=0. Conversely, for any r∈R, we have e⁢r=e⁢r-e⁢r⁢f=e⁢r⁢(1-f)=e⁢r⁢e=(1-f)⁢r⁢e=r⁢e-f⁢r⁢e=r⁢e. ∎

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 ring.

Proof.

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

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

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

      e⊕(f⊕g)	=	e+(f+g-f⁢g)-e⁢(f+g-f⁢g)
      =	e+f+g-e⁢f-f⁢g-e⁢g+e⁢f⁢g

   | = | (e+f-e⁢f)+g-(e+f-e⁢f)⁢g | |
   | = | (e⊕f)⊕g. | |

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

      e⊙(f⊕g)	=	e⁢(f+g-f⁢g)=e⁢f+e⁢g-e⁢f⁢g
      =	e⁢f+e⁢g-e⁢e⁢f⁢g=e⁢f+e⁢g-e⁢f⁢e⁢g

   | = | e⁢f⊕e⁢g=(e⊙f)⊕(e⊙g). | |

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