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Operations on a ring

Source: Romanian District Olympiad 2014, Grade 12, P3

June 15, 2014
superior algebrasuperior algebra unsolved

Problem Statement

Let (A,+,⋅)(A,+,\cdot) be an unit ring with the property: for all x∈Ax\in A, x+x2+x3=x4+x5+x6 x+x^{2}+x^{3}=x^{4}+x^{5}+x^{6} [*]Let x∈Ax\in A and let n≥2n\geq2 be an integer such that xn=0x^{n}=0. Prove that x=0x=0. [*]Prove that x4=xx^{4}=x, for all x∈Ax\in A.
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