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

Abstract

Near-rings considered are right near-rings and R is a near-ring. Jr 0 (R), the right Jacobson radical of R of type-0, was introduced and studied by the present authors. In this paper Jr 1 (R) and Jr 2 (R), the right Jacobson radicals of R of type-1 and type-2 are introduced. It is proved that both Jr 1 and Jr 2 are radicals for near-rings and Jr 0 (R) µ Jr 1 (R) µ Jr 2 (R). Unlike the left Jacobson radical classes, the right Jacobson radical class of type-2 contains M0(G) for many of the finite groups G. Depending on the structure of G, M0(G) belongs to dierent 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 Ri, i 2 I, Jr º (©i2IRi) = ©i2IJr º (Ri), º 2 f1; 2g. Moreover, under certain conditions, for an invariant subnear-ring S of a d.g. near-ring R it is shown that Jr(S) = S Jr(R).