A note on primitive distributively generated near-rings
Right near-rings with identity 1 are considered. Let \(N\) be such a near- ring. A right \(N\)-module is a group \((M,+)\), on which \(N\) acts from the right, such that the following conditions are satisfied: (i) For all \(n\in N\) the mapping \(m \to mn\) is an endomorphism of the group \((M,+)\); (ii) \(m(n_ 1n_ 2) = (mn_ 1)n_ 2\), (iii) \(m \cdot 1 = m\), (iv) \(m\cdot 0_ N = 0_ M\), for all \(m \in M\), all \(n_ i \in N\). The near-ring \(N\) is called \(d\)-primitive of type 2, if there exists a faithful right \(N\)-module \(M\neq (0)\) with the properties (i) The group \((M,+)\) has a set \(S\) of generators, such that \(s(n_ 1 + n_ 2) = sn_ 1 + sn_ 2\) for all \(s \in S\), all \(n_ i \in N\). (ii) \(M\) has no proper \(N\)-subgroups. Finally, \(N\) is called \(d\)-primitive of type 1, if there exists a faithful right \(N\)-module \(M\neq (0)\) with the properties (i) as in the type 2 case; (ii) \(M\) has no proper ``cyclic'' \(N\)- subgroups. The purpose of this paper is to establish a density theorem for \(d\)- primitive near-rings of type 1, and its consequences under the DCC for right ideals (Notabene: The author considers right near-rings). The proofs are to some extent adaptations of arguments from the reviewer's paper [in Math. Z. 130, 351-361 (1973; Zbl 0267.16017)]. Reviewer's remark: At various places the author fails to list all relevant assumptions; hence the paper should be quoted with some caution. E.g. the density theorem 2.4 is proved under the assumption that the near-ring in question is not a ring. Etc.
scientific article; zbMATH DE number 404417