Kurosh–Amitsur Right Jacobson Radicals of Type-1 and 2 for Right Near-Rings

Abstract

Near-rings considered are right near-rings. Let ν ∈ {1, 2}. J r ν , the right Jacobson radical of type-ν, was introduced for near-rings by the first and second authors. In this paper properties of these radicals J r ν are studied. It is shown that J r ν is a Kurosh-Amitsur radical (KA-radical) in the variety of all near-rings R in which the constant part Rc of R is an ideal of R. Thus, unlike the left Jacobson radical of type-1 of near-rings, J r 1 is a KA-radical in the class of all zero-symmetric near-rings. J r ν is not s-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings. Mathematics Subject Classification (2000). Primary 16Y30.Keywords. Right R-groups of type-1 and 2, right primitive ideals of type-1 and 2, right Jacobson radicals of type-1 and 2.

Hereditary right Jacobson radicals of type-1(e) and 2(e) for right near-rings

“…A proof similar to the one given for Proposition 3.21 of [13] works here also, which uses Corollary 3.17. Proof.…”

“…We begin with some basic properties of right R-groups of type-ν. The following Proposition is proved in [11] (Corollary 3.4). We give here a different proof.…”

“…Now we give an example of a right R-group of type-ν which is not of typeν(e). This example was considered in [3] and [13]. Proof.…”

“…Let K be a right ideal of R. Then the ideal {0} of R is contained in K. Since K is a subgroup of (R, +) if I and J are ideals of R contained in K, then I + J ⊆ K. So there is a largest ideal of R contained in K.The following Proposition follows from Proposition 3.19 of[13].Proposition 3.16. Let G be right R-group of type-ν(e) and P := (0 : G) = {r ∈ R | Gr = {0}}.…”

“…theorem of rings involving the matrix rings to near-rings, is established in [12]. In [10] and [11] the authors with T. Srinivas have shown that the right Jacobson radicals of type-0, 1 and 2 are Kurosh-Amitsur radicals (KA-radicals) in the class of all zero-symmetric near-rings but they are not ideal-hereditary in that class. In this paper right R-groups of type-ν(e), right ν(e)-primitive ideals and right ν(e)-primitive near-rings are introduced, ν ∈ {1, 2}.…”