A KUROSH-AMITSUR LEFT JACOBSON RADICAL FOR RIGHT NEAR-RINGS

Abstract

Let R be a right near-ring. An R-group of type-5/2 which is a natural generalization of an irreducible (ring) module is introduced in near-rings. An R-group of type-5/2 is an R-group of type-2 and an R-group of type-3 is an R-group of type-5/2. Using it <TEX>$J_{5/2}$</TEX>, the Jacobson radical of type-5/2, is introduced in near-rings and it is observed that <TEX>$J_2(R){\subseteq}J_{5/2}(R){\subseteq}J_3(R)$</TEX>. It is shown that <TEX>$J_{5/2}$</TEX> is an ideal-hereditary Kurosh-Amitsur radical (KA-radical) in the class of all zero-symmetric near-rings. But <TEX>$J_{5/2}$</TEX> is not a KA-radical in the class of all near-rings. By introducing an R-group of type-(5/2)(0) it is shown that <TEX>$J_{(5/2)(0)}$</TEX>, the corresponding Jacobson radical of type-(5/2)(0), is a KA-radical in the class of all near-rings which extends the radical <TEX>$J_{5/2}$</TEX> of zero-symmetric near-rings to the class of all near-rings.