MathDB

18

Part of 1982 IMO Longlists

Problems(1)

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

Source: IMO LongList 1982 - P18

3/16/2011
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.
abstract algebraalgebra unsolvedalgebra