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If $a,b,x$ in any ring, such that $a+(1−ab)x$ is a unit. Prove that there is a $y$ in this ring such that $b+y(1−ab)$ is a unit.

I am trying to find $r_1 , r_2 , r_3$ in the ring that $r_1(b+r_2(1−ab))r_3$ is equal to $a+(1−ab)x$ , so I can prove that there is an ideal $(b+r_2(1−ab))=R$

Anne Bauval
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Dual Rray
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  • (I did not downvote) A hint from Lam's A First Course in Noncommutative Rings, Exercise 1.25 ($c = 1-ab$): Set $u = a + cx \in U(R)$, and check that the element $y = (1 - bx)u^{-1}$ works. For this choice of $y$, an inverse for $b + yc$ is given by $a + x(1 - ba)$. The calculations are tricky, and have to be carried out carefully. – Amateur_Algebraist Oct 05 '24 at 10:38
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    I solved the problem, thanks for the tip.Without this hint, how can I solve it on my own?Because I feel like I can't think of the solution? – Dual Rray Oct 05 '24 at 11:13
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    See also this famous problem of Halmos. $\ \ $ – Bill Dubuque Oct 05 '24 at 17:52
  • I'm afraid I'm not sure how... This proof apparently first appears in Goodearl, Cancellation of low-rank vector bundles (1984), Lemma 3.1. – Amateur_Algebraist Oct 05 '24 at 17:58

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$\left(b+yc\right)\left(a+x\left(1-ba\right)\right)\\ =ba+bx(1-ba)+yca+ycx(1-ba)\\ =ba+bx-bxba+u^{-1}(a+x-abx)-bxu^{-1}(a+x-abx)-u^{-1}(a+x-abx)ba+bxu^{-1}(a+x-abx)ba\\ =ba+bx-bxba+1-bx-ba+bxba\\ =1$

Dual Rray
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  • Please copy-paste Lam's definitions of $c,u,y$ (provided by @Amateur_Algebraist's comment above) in your self-answer. – Anne Bauval Oct 05 '24 at 12:42