idempotent

An element $x$ of a ring is called an idempotent element, or simply an idempotent if $x^2=x$.

The set of idempotents of a ring can be partially ordered by putting $e\le f$ iff $e=ef=fe$.

The element $0$ is a minimum element in this partial order. If the ring has an identity element, $1$, then $1$ is a maximum element in this partial order.

Since the above definitions refer only to the multiplicative structure of the ring, they also hold for semigroups (with the proviso, of course, that a semigroup may have neither a zero element nor an identity element). In the special case of a semilattice, this partial order is the same as the one described in the entry for semilattice.

If a ring has an identity, then $1-e$ is always an idempotent whenever $e$ is an idempotent, and $e(1-e)=(1-e)e=0$.

In a ring with an identity, two idempotents $e$ and $f$ are called a pair of orthogonal idempotents if $e+f=1$, and $ef=fe=0$. Obviously, this is just a fancy way of saying that $f=1-e$.

More generally, a set $\{e_1,e_2,\mathrm{\dots },e_n\}$ of idempotents is called a complete set of orthogonal idempotents if $e_i{e}_j=e_j{e}_i=0$ whenever $i\ne j$ and if $1=e_1+e_2+\mathrm{\dots }+e_n$.

If $\{e_1,e_2,\mathrm{\dots },e_n\}$ is a complete set of orthogonal idempotents,and in addition each $e_i$ is in the centre of $R$, then each $R{e}_i$ is a subring, and

$$R\cong R{e}_1\times R{e}_2\times \mathrm{\dots }\times R{e}_n.$$

Conversely, whenever $R_1\times R_2\times \mathrm{\dots }\times R_n$ is a directproduct of rings with identities, write $e_i$ for the element of the productcorresponding to the identity element of $R_i$. Then $\{e_1,e_2,\mathrm{\dots },e_n\}$ is a complete set of central orthogonal idempotents of the product ring.

When a complete set of orthogonal idempotents is not central, there is a more complicated : see the entry on the Peirce decomposition for the details.

Title	idempotent
Canonical name	Idempotent
Date of creation	2013-03-22 13:07:27
Last modified on	2013-03-22 13:07:27
Owner	mclase (549)
Last modified by	mclase (549)
Numerical id	11
Author	mclase (549)
Entry type	Definition
Classification	msc 16U99
Classification	msc 20M99
Synonym	idempotent element
Related topic	Semilattice
Related topic	Idempotency
Defines	orthogonal idempotents
Defines	complete set of orthogonal idempotents