Let R be a general ring, not necessarily commutative. An element x∈R is said to be idempotent if x2 = x. Note An endomorphism f of an R-module M (i.e. ) is an idempotent if and only if fis a projection, i.e. M = ker(f) ⊕ im(f) and f : M→M projects onto im(f). Indeed ⇐ is obvious, and conversely if f is idempotent, we have:  Every m∈M is just f(m) + (m – f(m)). The first term is in im(f); the second term lies in ker(f) since f(m – f(m)) = f(m) – f2(m) = f(m) – f(m) = 0. So M = ker(f) + im(f).  Any element of ker(f) ∩ im(f) can be written as f(m) such that f(f(m)) = 0. But this means f(m) = f2(m) = 0, so ker(f) ∩ im(f) = 0. Throughout this article, we shall focus on idempotents which commute, i.e. ef = fe. A set of idempotents {ei} is said to be orthogonal if eiej = 0 for all i≠j. The following are easy to prove. 1. The sum of two orthogonal idempotents is also an idempotent. 2. If e is any idempotent, then e and 1-e are orthogonal idempotents.