Strongly monogenic near-rings
Let N be a left zero-symmetric near-ring. N is said to be strongly monogenic (s.m.) if aN\(\in \{0,N\}\) for all \(a\in N\) and \(bN=N\) for some \(b\in N\). N is said to be geometric if for each a,b\(\in N\) there is an \(e\in N\) such that \(ea=a\) and if ab\(\neq 0\) then \(Nab=Nb\). It is shown that an s.m. near-ring N has a unique maximal ideal R, equal to the Jacobson-type radical \(J_ 1(N)\) and to the prime radical P(N). Moreover \(J_ 2(N)\) is equal either to R or to N. Some characterizations of near- rings that are both s.m. and geometric are also given.
scientific article; zbMATH DE number 4023514