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idempotent element
In abstract algebra, an element ''x'' of a set with a binary operation ∗ is called an idempotent element (or just an idempotent) if . This reflects the idempotence of the binary operation on that particular element.
Idempotents are especially prominent in ring theory. For general rings, elements idempotent under multiplication are tied with decompositions of modules, as well as to homological properties of the ring. In Boolean algebra, the main objects of study are rings in which all elements are idempotent under both addition and multiplication.
==Definitions==
An idempotent element of a ring is an element ''a'' such that .〔See Hazewinkel et al. (2004), p. 2.〕 That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that for any positive integer ''n''.
There are many special types of idempotents defined after the following examples section.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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