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ring problem

Source: Romania District Olympiad 2013,grade XII(problem 4)

March 14, 2013
superior algebrasuperior algebra unsolved

Problem Statement

Problem 4. Let(A,+,⋅)\left( A,+,\cdot \right) be a ring with the property that x=0x=0 is the only solution of the x2=0,x∈A{{x}^{2}}=0,x\in Aecuation. Let B={a∈A∣a2=1}B=\left\{ a\in A|{{a}^{2}}=1 \right\}. Prove that: (a) ab−ba=bab−aab-ba=bab-a, whatever would be a∈Aa\in A and b∈Bb\in B. (b) (B,⋅)\left( B,\cdot \right) is a group
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