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# idempotent

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

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 decomposition: see the entry on the Peirce decomposition for the details.

## Mathematics Subject Classification

16U99*no label found*20M99

*no label found*

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## Comments

## Test

Does L have a Siegel zero? If that's the case, is an effective version necessary?