# Answer on Algebra Question for Tsit Lam

Question #16536

Let a, b be elements in a ring R. If 1 − ba is left-invertible, show that 1 − ab is left-invertible, and construct a left inverse for it explicitly

Expert's answer

The left ideal

cases

*R*(1*− ab*) contains*Rb*(1*− ab*) =*R*(1*− ba*)*b*=*Rb,*so it also contains (1*− ab*) +*ab*= 1. This shows that 1*− ab*is left-invertible. This proof lends itself easily to an explicit construction: if*u*(1*− ba*) = 1,then*b*=*u*(1*− ba*)*b*=*ub*(1*− ab*)*,*so 1 = 1*− ab*+*ab*= 1*− ab*+*aub*(1*− ab*) = (1+*aub*)(1*− ab*)*.*

Hence, (1*− ab*)*−*1 = 1+*a*(1*− ba*)*−*1*b,*where*x−*1 denotes “a left inverse” of*x*. The case when 1*− ba*is invertible follows by combining the “left-invertible” and “right-invertible”cases

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