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# Answer to Question #16536 in Algebra 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 R(1 &minus; ab) contains Rb(1 &minus; ab) = R(1 &minus; ba)b = Rb, so it also contains (1 &minus; ab) + ab = 1. This shows that 1 &minus; ab is left-invertible. This proof lends itself easily to an explicit construction: if u(1 &minus; ba) = 1,then
b = u(1 &minus; ba)b = ub(1 &minus; ab), so 1 = 1&minus; ab + ab = 1&minus; ab + aub(1 &minus; ab) = (1+aub)(1 &minus; ab).
Hence, (1 &minus; ab)&minus;1 = 1+a(1 &minus; ba)&minus;1b, where x&minus;1 denotes &ldquo;a left inverse&rdquo; of x. The case when 1 &minus; ba is invertible follows by combining the &ldquo;left-invertible&rdquo; and &ldquo;right-invertible&rdquo;
cases

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