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Finite ring with unique solution

Source: RMO 2008, Grade 12, Problem 3

April 30, 2008
Ring Theorylinear algebramatrixnumber theoryleast common multiplesearchgroup theory

Problem Statement

Let A A be a unitary finite ring with n n elements, such that the equation x^n\equal{}1 has a unique solution in A A, x\equal{}1. Prove that a) 0 0 is the only nilpotent element of A A; b) there exists an integer k≥2 k\geq 2, such that the equation x^k\equal{}x has n n solutions in A A.
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