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(a + ab^(-1)a)^(-1) + (a + b)^(-1) = a^(-1) in every ring

Source: IMO LongList 1982 - P18

March 16, 2011
abstract algebraalgebra unsolvedalgebra

Problem Statement

You are given an algebraic system admitting addition and multiplication for which all the laws of ordinary arithmetic are valid except commutativity of multiplication. Show that (a+ab−1a)−1+(a+b)−1=a−1,(a + ab^{-1} a)^{-1}+ (a + b)^{-1} = a^{-1}, where x−1x^{-1} is the element for which x−1x=xx−1=ex^{-1}x = xx^{-1} = e, where ee is the element of the system such that for all aa the equality ea=ae=aea = ae = a holds.
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