Solving some linear equations over alternative rings (Q606396)

A (not necessarily associative) ring \(R\) is said to be alternative if the multiplication in \(R\) satisfies the ``alternative'' laws: \(a(ab)=(a^2)b;~(ba)a=b(a^2)\) for all \(a,b \in R\). Let \(R\) be an alternative ring. The author of the paper under review considers the solvability of the linear equations \(ax=b,xa=b\) and \(axa=b\) for all \(a,b \in R\). Necessary and sufficient conditions for the existence of each of the equations is given. These results are also supplemented with the expression for the most general solution and conditions guaranteeing uniqueness of solutions. Each of these conditions involve the von Neumann regularity of the element \(a\). (\(a \in R\) is said to be von Neumann regular if there exists \(x \in R\) such that \(axa=a\)). Further, the author derive conditions under which each of the equations \(ax-ya=b,ax-xa=b\) and \(x-axa=b\) have solutions in \(R\), where \(a\) is assumed to be either idempotent or involutory.