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Concept
Idempotent (ring theory)
Summary
In ring theory, a branch of mathematics, an idempotent element or simply 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.
One may consider the ring of integers modulo n where n is squarefree. By the Chinese remainder theorem, this ring factors into the product of rings of integers modulo p where p is prime. Now each of these factors is a field, so it is clear that the factors' only idempotents will be 0 and 1. That is, each factor has two idempotents. So if there are m factors, there will be 2m idempotents.
We can check this for the integers mod 6, R = Z/6Z. Since 6 has two prime factors (2 and 3) it should have 22 idempotents.
02 ≡ 0 ≡ 0 (mod 6)
12 ≡ 1 ≡ 1 (mod 6)
22 ≡ 4 ≡ 4 (mod 6)
32 ≡ 9 ≡ 3 (mod 6)
42 ≡ 16 ≡ 4 (mod 6)
52 ≡ 25 ≡ 1 (mod 6)
From these computations, 0, 1, 3, and 4 are idempotents of this ring, while 2 and 5 are not. This also demonstrates the decomposition properties described below: because 3 + 4 = 1 (mod 6), there is a ring decomposition 3Z/6Z ⊕ 4Z/6Z. In 3Z/6Z the identity is 3+6Z and in 4Z/6Z the identity is 4+6Z.
Given a ring and an element such that , then the quotient ring
has the idempotent . For example, this could be applied to , or any polynomial .
There is a catenoid of idempotents in the split-quaternion ring.
A partial list of important types of idempotents includes:
Two idempotents a and b are called orthogonal if ab = ba = 0. If a is idempotent in the ring R (with unity), then so is b = 1 − a; moreover, a and b are orthogonal.
Official source
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