central idempotent
Let $R$ be a ring. An element $e\in R$ is called a central idempotent if it is an idempotent^{} and is in the center $Z(R)$ of $R$.
It is wellknown that if $e\in R$ is an idempotent, then $eRe$ has the structure^{} 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 operation^{} 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 $f:=1e$ 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 $\mathrm{1}$ is central iff $e\mathit{}R\mathit{}f\mathrm{=}f\mathit{}R\mathit{}e\mathrm{=}\mathrm{0}$, where $f\mathrm{=}\mathrm{1}\mathrm{}e$.
Proof.
If $e$ is central, then clearly $eRf=fRe=0$. Conversely, for any $r\in R$, we have $er=ererf=er(1f)=ere=(1f)re=refre=re$. ∎
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 $\mathrm{1}$ has the structure of a Boolean ring^{}.
Proof.
First, note that $0,1\in C$. Next, for $e,f\in C$, we define addition $\oplus $ and multiplication $\odot $ on $C$ as follows:
$$e\oplus f:=e+fef\mathit{\hspace{1em}\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em}}e\odot f:=ef.$$ 
As discussed above, $\oplus $ and $\odot $ are welldefined (as $C$ is closed under these operations). In addition, for any $e,f,g\in C$, we have

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

2.
$\oplus $ is commutative^{}, since $C\subseteq Z(R)$.

3.
$\oplus $ is associative:
$e\oplus (f\oplus g)$ $=$ $e+(f+gfg)e(f+gfg)$ $=$ $e+f+geffgeg+efg$ $=$ $(e+fef)+g(e+fef)g$ $=$ $(e\oplus f)\oplus g.$ 
4.
$\odot $ distributes over $\oplus $: we only need to show left distributivity (since $\odot $ is commutative by $1$ above):
$e\odot (f\oplus g)$ $=$ $e(f+gfg)=ef+egefg$ $=$ $ef+egeefg=ef+egefeg$ $=$ $ef\oplus eg=(e\odot f)\oplus (e\odot g).$
This shows that $(C,0,1,\oplus ,\odot )$ is a Boolean ring. ∎
Title  central idempotent 

Canonical name  CentralIdempotent 
Date of creation  20130322 19:13:07 
Last modified on  20130322 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 