Solving Some Linear Equations Over Alternative Rings

One of the fundamental research problems in algebra theory is to solve equations over the given algebraic structures. The purpose of this note is to solve some basic linear equations over alternative rings. In particular, we derive necessary and sufficient conditions for the equations ax = b, xa = b and axa = b to be solvable over an alternative ring, respectively, and give general solutions of these equations. In addition, we solve the three linear equations ax ? ya = b, ax ? xa = b and x ? axa = b over alternative rings, respectively, where the coefficient element a is idempotent or involutory.