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idempotents
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| Plural form of idempotent. |
| Denoting an element of a set which is unchanged in value when multiplied or otherwise operated on by itself. |
| An idempotent element. |
| Idempotents might refer to |
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In ring theory (part of abstract algebra) an idempotent element, or simply an idempotent, of a ring is an element a such that a2 = a. That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that a = a2 = a3 = a4 = ... = an 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 involved in decompositions of modules, and connected 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. |