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If in a ring R, 1 - ab is a unit, will 1-ba be a unit?

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FACEBOOK Problem Q.: If in a ring R, 1 ab is a unit, will 1 ba be a unit? Solution: The answer is YES and the proof is given as follows: Since 1 ab is a unit, so c R such that c(1 ab) = 1 = (1 ab)c. Claim: (1 ba)(1 + bca) = 1 = (1 + bca)(1 ba). We have, (1 ba)(1 + bca) = 1 + bca ba babca = 1 ba + bca babca = 1 ba + b(1 ab)ca = 1 ba + ba = 1. [ since (1 ab)c = 1 ] Similarly, (1 + bca)(1 ba) = 1. Hence, (1 ba)(1 + bca) = 1 = (1 + bca)(1 ba) and so 1 ba is a unit in R. Solved by Arnab Bhattacharjee. 1

 

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