An example of a projective near-field with non-projective sub-near-fields
A projective near-field is a near-field \(F\) which satisfies the property that for all \(a,b,c\in F\) with \(a\neq b\), there exists an \(x\in F\) such that \(ax - bx = c\). Projective near-fields were defined by \textit{W. Kerby} [Abh. Math. Semin. Univ. Hamb. 32, 20--24 (1968; Zbl 0182.53702)]. The term projective indicates the relationship between projective near-fields and projective planes, for if a near-field is projective, it is ternary ring, and hence it determines a projective plane. The purpose of this note is to give an example of a projective near-field which contains a non-projective sub-near-field. The example is constructed by defining a new multiplication an the ring of formal power series over a polynomial ring. This near-field was first studied by \textit{F. Pokropp} [Abh. Math. Semin. Univ. Hamb. 30, 188--219 (1967; Zbl 0147.27901)] and it was shown to be projective by W. Kerby [loc. cit.]. The present author shows that the sub-near-field of rational functions over the polynomial ring is not projective.
scientific article; zbMATH DE number 3557905