Peirce decomposition

Let $e$ be an idempotent of a ring $R$, not necessarily with an identity.For any subset $X$ of $R$, we introduce the notations:

$$(1-e)X=\{x-ex\mid x\in X\}$$

and

$$X(1-e)=\{x-xe\mid x\in X\}.$$

If it happens that $R$ has an identity element, then $1-e$ is a legitimate element of$R$, and this notation agrees with the usual product of an element and a set.

It is easy to see that $Xe\cap X(1-e)=0=eX\cap (1-e)X$ for any set $X$ which contains $0$.

Applying this first on the right with $X=R$ and then on the left with $X=Re$ and $X=R(1-e)$,we obtain:

$$R=eRe\oplus eR(1-e)\oplus (1-e)Re\oplus (1-e)R(1-e).$$

This is called the Peirce Decompostion of $R$ with respect to $e$.

Note that $eRe$ and $(1-e)R(1-e)$ are subrings, $eR(1-e)$ is an $eRe$-$(1-e)R(1-e)$-bimodule,and $(1-e)Re$ is a $(1-e)R(1-e)$-$eRe$-bimodule.

This is an example of a generalized matrix ring:

More generally, if $R$ has an identity element,and $e_1,e_2,\mathrm{\dots },e_n$ is a complete set of orthogonal idempotents,then

is a generalized matrix ring.