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Answer to Question #17265 in Abstract Algebra for Tsit Lam

Question #17265
Show that, if R has an identity 1, the map ϕ : (R, ◦) → (R,×) sending a to 1 − a is a monoid isomorphism. In this case, an element a is left (right) quasi-regular iff 1 − a has a left (resp. right) inverse with respect to ring multiplication.
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