Idempotent element (ring theory)
In abstract algebra, more specifically in ring theory, 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 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.
One may consider the ring of integers mod n, where n is squarefree. By the Chinese remainder theorem, this ring factors into the direct product of rings of integers mod p. Now each of these factors is a field, so it is clear that the 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 factors (2 and 3) it should have 22 idempotents.
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.
There is a catenoid of idempotents in the split-quaternion ring.
A partial list of important types of idempotents includes:
Any non-trivial idempotent a is a zero divisor (because ab = 0 with neither a nor b being zero, where b = 1 − a). This shows that integral domains and division rings do not have such idempotents. Local rings also do not have such idempotents, but for a different reason. The only idempotent contained in the Jacobson radical of a ring is 0.
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