Kurosh-Amitsur right Jacobson radicals of type-1 and 2 for right near-rings. (Q927816)

Earlier the first two authors defined and investigated the right Jacobson radicals \(J_1^r\) and \(J_2^r\) for right near-rings [\textit{R. Srinivasa Rao} and \textit{K. Siva Prasad}, Kyungpook Math. J. 46, No. 4, 603-613 (2006; Zbl 1122.16037)]. Further properties of these two radicals which are of interest from a general radical theoretic view are determined here. In particular, it is shown that \(J_\nu^r\), \(\nu\in\{1,2\}\), is a Kurosh-Amitsur radical in the variety of all near-rings in which the constant part forms an ideal. This means \(J_1^r\) is a Kurosh-Amitsur radical in the variety of zero-symmetric near-rings which is not the case for the left Jacobson radical \(J_1\). However, examples are provided to show that for each \(\nu\in\{1,2\}\), \(J_\nu^r\) is not s-hereditary (and thus not ideal-hereditary) in the variety of zero-symmetric near-rings.