Question

For any ring R, show that the following are equivalent:
(1) For any a ∈ R, there exists a unit u ∈ U(R) such that a = aua.
(2) Every a ∈ R can be written as a unit times an idempotent.
(2’) Every a ∈ R can be written as an idempotent times a unit.

Answer 1

By left-right symmetry, it sufficesto prove (1) ⇐⇒ (2).
(1) ⇒ (2). Write a = aua where u ∈ U(R). If e : = ua, then
e² = uaua = ua = e, and a= u^−1e, as desired.
(2) ⇒ (1). Given a ∈ R, write a= ve where v ∈ U(R) and e²= e. The latter implies v^−1a = v^−1av^−1a,so a = av^−1a, as desired.