# Idempotent

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idempotent element

An element $e$ of a ring, semi-group or groupoid equal to its own square: $e^2=e$. An idempotent $e$ is said to contain an idempotent $f$ (denoted by $e\geq f$) if $ef=e=fe$. For associative rings and semi-groups, the relation $\geq$ is a partial order on the set $E$ of idempotent elements, called the natural partial order on $E$. Two idempotents $u$ and $v$ of a ring are said to be orthogonal if $uv=0=vu$. With every idempotent of a ring (and also with every system of orthogonal idempotents) there is associated the so-called Peirce decomposition of the ring. For an $n$-ary algebraic relation $\omega$, an element $e$ is said to be an idempotent if $(e\cdots e)\omega=e$, where $e$ occurs $n$ times between the brackets.

An algebraic operation $\omega$ is sometimes said to be idempotent if every element of the set on which it acts is idempotent in the sense defined above. Such operations are also called affine operations; the latter name is preferable because an affine unary operation is not the same thing as an idempotent element of the semi-group of unary operations. In the theory of $R$-modules, the affine operations are those of the form
$$(x_1,\dotsc,x_n)\mapsto\sum_{i=1}^nr_ix_i$$
with $\sum_{i=1}^nr_i=1$.