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 well-known 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 $f:=1-e$ 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 $\mathrm{1}$ 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 $er=er-erf=er(1-f)=ere=(1-f)re=re-fre=re$. ∎

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 $\mathrm{1}$ 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:

$$e\oplus f:=e+f-ef\mathit{\hspace{1em}\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em}}e\odot f:=ef.$$

As discussed above, $\oplus$ and $\odot$ are well-defined (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:

   $e\oplus (f\oplus g)$ $=$ $e+(f+g-fg)-e(f+g-fg)$

   $=$ $e+f+g-ef-fg-eg+efg$

   $=$ $(e+f-ef)+g-(e+f-ef)g$

   $=$ $(e\oplus f)\oplus g.$

4. $\odot$ distributes over $\oplus$: we only need to show left distributivity (since $\odot$ is commutative by $1$ above):

   $e\odot (f\oplus g)$ $=$ $e(f+g-fg)=ef+eg-efg$

   $=$ $ef+eg-eefg=ef+eg-efeg$

   $=$ $ef\oplus eg=(e\odot f)\oplus (e\odot g).$

This shows that $(C,0,1,\oplus ,\odot )$ is a Boolean ring. ∎

Title	central idempotent
Canonical name	CentralIdempotent
Date of creation	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