It is easy to prove that if every element of a ring is idempotent, then the ring is commutative. This fact can be generalized as follows.

**Problem**. 1) Let be a ring with identity and suppose that every element of is a product of idempotent elements. Prove that is commutative.

2) Give an example of a noncommutative ring with identity such that every element of is a product of some elements of the set

**Solution**. 1) Obviously we only need to prove that every idempotent is central. Suppose first that for some We claim that So suppose the claim is false. Then where are idempotents and Let Then and hence Thus Contradiction! Now suppose that for some Then and therefore , by what we just proved. Finally, since for any idempotent and any we have and so is central.

2) One example is the ring of upper triangular matrices with entries from