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 *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”

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