Idempotent elements in a ring

Let R be a commutative ring with 1 and suppose e∈R is idempotent, i.e., satisfies e2=e.

  1. Prove that 1βˆ’e is also idempotent.
  2. Suppose eβ‰ 0,1. Show that Re and R(1βˆ’e) are proper ideals of R.
  3. Prove there is an isomorphism Rβ‰…ReΓ—R(1βˆ’e).