# 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 direct
product^{} of rings with identities, write ${e}_{i}$ for the element of the product
corresponding to the identity element of ${R}_{i}$. Then $\{{e}_{1},{e}_{2},\mathrm{\dots},{e}_{n}\}$ 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 |