If a polynomial identity guarantees that every partial order on a ring can be extended, then this identity is true only for a zero-ring
A ring \(R\) is a zero-ring if \(xy = 0\) for all \(x,y\) in \(R\), and is called an \(O\)-ring if it can be linearly ordered (so that the linear order is compatible with the ring + and \(\times \}\). An \(O\)-ring \(R\) is called an \(O^*\)-ring if every partial order on \(R\) can be extended to a linear order. It is well known that if an abelian group \(G\) can be linearly ordered, then every partial order on \(G\) can be extended to a linear order. \textit{L. Fuchs} [Partially ordered algebraic systems (1963; Zbl 0137.020)] asked if there exists a polynomial identity which implies that an \(O\)-ring must be an \(O^*\)-ring. In the case for groups, \(xy = yx\) is such an identity. The author proves that no such identity is possible, namely that every ring \(R\) that satisfies such an identity is a zero- ring.
scientific article; zbMATH DE number 750793