Jordan structure in semirings (Q6054087)

The authors study prime semirings and their commutativity. As is the case for rings, important and useful tools for such investigations are the Jordan ideals, the Lie ideals as well as derivations. These notions are defined for semirings analogous to their ring theory counterparts. Subsequently many of the ring theory results are generalized to semirings. These generalizations are facilitated by a ring defined earlier by the authors, namely the ring of differences of a semiring induced by a derivation on the semiring. Two of the main results are: (1) Let \(R\) be an additively cancellative yoked prime semiring and let \(d\) be a derivation of \(R\). Let \(U\) be a Jordan ideal and a subsemiring of \(R\) such that \(ud(u)=d(u)u\) for all \(u\in U\). Let \(R^{\vartriangle }\) be the ring of differences of \(R\), \(d^{\vartriangle }\) the corresponding derivation of \(d\) and \(U^{\vartriangle }\) the corresponding Jordan ideal of \( U\). If \(R\) is of characteristic two, then \(U\) is commutative. (2) Let \(R\) be be an additively cancellative yoked semiprime semiring and let \(R^{\vartriangle }\) be its corresponding ring of differences. If \(d\) is a Jordan derivation of \(R\) and if \(R^{\vartriangle }\) is \(2\)-torsion free, then \(d\) is just an ordinary derivation of \(R\).