What is "idempotent"

1870, from Latin idem "the same" + potentem "powerful" (see potent).

a. 1 (context mathematics computing English) Describing an action which, when performed multiple times, has no further effect on its subject after the first time it is performed. 2 (context mathematics English) ''Said of an element of an algebraic structure (such as a group or semigroup) with a binary operation'': that when the element operates on itself, the result is equal to itself. 3 (context mathematics English) ''Said of a binary operation'': that all of the distinct elements it can operate on are idempotent (in the sense given just above). 4 (context mathematics English) ''Said of an algebraic structure'': having an idempotent operation (in the sense above). n. 1 (context mathematics English) An idempotent element. 2 (context mathematics English) An idempotent structure.

adj. unchanged in value following multiplication by itself; "this matrix is idempotent"

In abstract algebra, more specifically in ring theory, an idempotent element, or simply an idempotent, of a ring is an element a such that . That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that for any positive integer n. For example, an idempotent element of a matrix ring is precisely an idempotent matrix.

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.