For a commutative ring an idempotent is an element such that . The collection is an ideal of and in fact, moreover, is a ring in its own right with identity (why?). A good example of an idempotent to have in mind is the element and in some sense, these are the only idempotents. is another commutative ring.
For idempotent we have also being an idempotent and . These should remind you of elements . We call a pairing such that and and complementary idempotents.
Complementary idempotents give a formulation of an inner product. What do I mean by this? If is a pair of complementary idempotents in , then defining , (remember these are rings in their own right) we have that where being a ring isomorphism. This is an instructive check to make and I leave it to the reader.
What do idempotents of some rings look like? is always idempotent, making the pair complementary idempotents. In all fields, the only non-zero idempotent element is the multiplicative unit .
What about rings ? if for some prime number , supposing that is an idempotent, we have and so if such that and if both then which is a contradiction, so or . The only idempotents are and . The only idempotent pair is .
We can now look at the general case, let be ‘s prime factorisation, the chinese remainder theorem tells us so there are idempotents, and pairs of complementary idempotents.