On near-fields (Q1078645)

This paper contains six necessary and sufficient conditions that certain (right) near-rings be near-fields. Some of these are: (i) A near-ring N is a near-field if and only if for every \(a\neq 0\) there exists a unique b in N such that \(aba=a\); (ii) let N be a near-ring in which idempotents commute, then N is a near-field if and only if for each \(a\neq 0\), \(Na=N\) and N0\(\neq N\); (iii) let N be a near-ring in which idempotents commute, then N is a near-field if and only if N is regular and N has no proper left ideals. The author indicates that some of his results are generalizations of theorems of \textit{S. Ligh} [Can. J. Math. 21, 1366-1371 (1969; Zbl 0191.029) and Math. Jap. 15, 7-13 (1970; Zbl 0209.335)].