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A condition for the commutativity of a unitary ring

Source: RMO 2004 - District Round

February 27, 2007
superior algebrasuperior algebra unsolved

Problem Statement

Let (A,+,⋅)\left( A,+, \cdot \right) be a ring that verifies the following properties: (i) it has a unit, 11, and its order is pp, a prime number; (ii) there is B⊂A, ∣B∣=pB \subset A, \, |B| = p, such that: for all x,y∈Ax,y \in A, there is b∈Bb \in B such that xy=byxxy = byx. Prove that AA is commutative. Ion Savu
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