Two More Radicals for Right Near-Rings: The Right Jacobson Radicals of Type-1 and 2

• Published in: Kyungpook mathematical journal Vol. 46; no. 4; pp. 603 - 613
• Main Authors: Rao, Ravi Srinivasa, Prasad, K. Siva
• Format: Journal Article
• Language: Korean
• Published: 경북대학교 자연과학대학 수학과 01.12.2006
• Subjects: 수학; right Jacobson radicals of type-1 and 2; right quasi-regular element; right 1-modular and 2-modular right ideals; near-ring
• ISSN: 1225-6951

Near-rings considered are right near-rings and R is a near-ring. $J_0^r(R)$, the right Jacobson radical of R of type-0, was introduced and studied by the present authors. In this paper $J_1^r(R)$ and $J_2^r(R)$, the right Jacobson radicals of R of type-1 and type-2 are introduced. It is proved that both $J_1^r$ and $J_2^r$ are radicals for near-rings and $J_0^r(R){\subseteq}J_1^r(R){\subseteq}J_2^r(R)$. Unlike the left Jacobson radical classes, the right Jacobson radical class of type-2 contains $M_0(G)$ for many of the finite groups G. Depending on the structure of G, $M_0(G)$ belongs to different right Jacobson radical classes of near-rings. Also unlike left Jacobson-type radicals, the constant part of R is contained in every right 1-modular (2-modular) right ideal of R. For any family of near-rings $R_i$, $i{\in}I$, $J_{\nu}^r({\oplus}_{i{\in}I}R_i)={\oplus}_{i{\in}I}J_{\nu}^r(R_i)$, ${\nu}{\in}\{1,2\}$. Moreover, under certain conditions, for an invariant subnear-ring S of a d.g. near-ring R it is shown that $J_2^r(S)=S{\cap}J_2^r(R)$.