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Answer to Question #17365 in Algebra for Melvin Henriksen

Question #17365
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.
Expert's answer
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.

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