# 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.

(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)

(1)

(2)

*⇐⇒**(2).*(1)

*⇒**(2). Write**a*=*aua*where*u**∈**U(**R*). If*e*: =*ua*, then*e*^{2}=*uaua*=*ua*=*e,*and*a*=*u*^{−}^{1}*e*, as desired.(2)

*⇒**(1). Given**a**∈**R*, write*a*=*ve*where*v**∈**U(**R*) and*e*^{2}=*e*. The latter implies*v*^{−}^{1}*a*=*v*^{−}^{1}*av*^{−}^{1}*a*,so*a*=*av*^{−}^{1}*a*, as desired.
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