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a sufficient condition for nilpotence

Source: Romanian District Olympiad 2015, Grade XII, Problem 4

September 26, 2018
Ring Theoryabstract algebranilpotencesuperior algebra

Problem Statement

Let m m be a non-negative ineger, n≥2 n\ge 2 be a natural number, A A be a ring which has exactly n n elements, and an element a a of A A such that 1−ak 1-a^k is invertible, for all k∈{m+1,m+2,...,m+n−1}. k\in\{ m+1,m+2,...,m+n-1\} . Prove that a a is nilpotent.
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