Regular duo near-rings
Let N be a near-ring. A subgroup Q of \((N,+)\) is called a quasi-ideal of the zero-symmetric near-ring N if QN\(\cap NQ\subseteq Q\). N is said to be a duo near-ring if every left or right N-subgroup is a 2-sided N-subgroup of N. A near-ring N is called an S-near-ring if \(n\in Nn\) for every element n in N. The main result of the author is Theorem 1: The following four conditions on a near-ring N are equivalent: (i) N is a regular duo near-ring; (ii) N is an S-near-ring and (*) \(AB=A\cap B\) holds for every couple of quasi-ideals of N; (iii) N is an S-near-ring and (*) holds for every couple of left N-subgroups and every couple of right N-subgroups of N; (iv) N is an S-near-ring and (*) holds for every left N-subgroup A and every right N-subgroup B of N. For analogous results concerning associative rings, see the reviewer [Proc. Japan Acad. 45, 157-158 (1969; Zbl 0185.087) and Ann. Univ. Sci. Budapest Rolando Eötvös, Sect. Math. 13, 71-72 (1970; Zbl 0211.061)] and \textit{O. Steinfeld} [Stud. Sci. Math. Hung. 8, 161-164 (1973; Zbl 0286.20085)].
scientific article; zbMATH DE number 4173185