# 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 $e\leq f$ 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 $\{e_{1},e_{2},\dots,e_{n}\}$ of idempotents is called a complete set of orthogonal idempotents if $e_{i}e_{j}=e_{j}e_{i}=0$ whenever $i\neq j$ and if $1=e_{1}+e_{2}+\dots+e_{n}$.

If $\{e_{1},e_{2},\dots,e_{n}\}$ is a complete set of orthogonal idempotents, and in addition each $e_{i}$ is in the centre of $R$, then each $Re_{i}$ is a subring, and

 $R\cong Re_{1}\times Re_{2}\times\dots\times Re_{n}.$

Conversely, whenever $R_{1}\times R_{2}\times\dots\times R_{n}$ is a direct product of rings with identities, write $e_{i}$ for the element of the product corresponding to the identity element of $R_{i}$. Then $\{e_{1},e_{2},\dots,e_{n}\}$ 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