Kurosh-Amitsur Right Jacobson Radical of Type 0 for Right Near-Rings

Abstract

By a near-ring we mean a right near-ring. 𝐽^𝑟₀, the right Jacobson radical of type 0, was introduced for near-rings by the first and second authors. In this paper properties of the radical 𝐽^𝑟₀ are studied. It is shown that 𝐽^𝑟₀ is a Kurosh-Amitsur radical (KA-radical) in the variety of all near-rings 𝑅, in which the constant part 𝑅_𝑐 of 𝑅 is an ideal of 𝑅. So unlike the left Jacobson radicals of types 0 and 1 of near-rings, 𝐽^𝑟₀ is a KA-radical in the class of all zero-symmetric near-rings. 𝐽^𝑟₀ is not 𝑠-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings.

1. Introduction

𝑅 denotes a right near-ring and all near-rings considered are right near-rings and not necessarily zero-symmetric.

In [1, 2], the first author studied the structure of near-rings in terms of right ideals, and showed that as in rings, matrix units determined by right ideals identify matrix near-rings. To show the importance of the right Jacobson radicals of near-rings in the extension of a form of the Wedderburn-Artin theorem of rings involving the matrix rings to near-rings, the right Jacobson radicals of type 𝜈 were introduced and studied by the first and second authors in [3–6], 𝜈∈{0,1,2,𝑠}. In [6], Wedderburn-Artin theorem was extended to near-rings, and some generalizations of it were presented.

In this paper, properties of the right Jacobson radical of type 0 are studied. It is known that the left Jacobson radicals of types 0 and 1 are not KA-radicals in the class of all zero-symmetric near-rings, and only the left Jacobson radicals of types 2 and 3 are KA-radicals in the class of all zero-symmetric near-rings. Surprisingly, 𝐽^𝑟₀, the right Jacobson radical of type 0, is a KA-radical in the class of all zero-symmetric near-rings. It is also shown that 𝐽^𝑟₀ is a KA-radical even in a bigger class of near-rings, namely, in the variety of all near-rings 𝑅, in which the constant part of 𝑅 is an ideal of 𝑅. Moreover, 𝐽^𝑟₀ is not 𝑠-hereditary, and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings.

2. Preliminaries

Near-rings considered are right near-rings and not necessarily zero-symmetric. Unless otherwise specified, 𝑅 stands for a right near-ring. Near-ring notions not defined here can be found in [7].

𝑅₀ and 𝑅_𝑐 denote the zero-symmetric part and the constant part of 𝑅, respectively.

ℱ denotes the class of near-rings 𝑅, in which the constant part 𝑅_𝑐 of 𝑅 is an ideal of 𝑅. In [8], Fuchs has shown that the class of near-rings ℱ is a variety. Obviously, ℱ contains all zero-symmetric, constant, and abstract affine near-rings. Now we give here some definitions and results of [3], which will be used later.

An element 𝑎∈𝑅 is called right quasiregular if and only if the right ideal of 𝑅 generated by the set {𝑥−𝑎𝑥∣𝑥∈𝑅} is 𝑅. A right ideal (left ideal, ideal, subset) 𝐾 of 𝑅 is called a right quasiregular right ideal (left ideal, ideal, subset) of 𝑅 if each element of 𝐾 is right quasiregular.

A right ideal 𝐾 of 𝑅 is called right modular if there is an element 𝑒∈𝑅 such that 𝑥−𝑒𝑥∈𝐾 for all 𝑥∈𝑅. In this case, we say that 𝐾 is right modular by 𝑒.

A maximal right modular right ideal of 𝑅 is called a right 0-modular right ideal of 𝑅.

𝐽^𝑟_1/2(𝑅) is the intersection of all right 0-modular right ideals of 𝑅, and if 𝑅 has no right 0-modular right ideals, then 𝐽^𝑟_1/2(𝑅) = 𝑅.

The largest ideal of 𝑅 contained in 𝐽^𝑟_1/2(𝑅) is denoted by 𝐽^𝑟₀(𝑅) and called the right Jacobson radical of 𝑅 of type 0.

The largest ideal contained in a right 0-modular right ideal of 𝑅 is called a right 0-primitive ideal of 𝑅. 𝑅 is called a right 0-primitive near-ring if {0} is a right 0-primitive ideal of 𝑅.

A group (𝐺, +) is called a right 𝑅-group if there is a mapping ((𝑔, 𝑟) →𝑔𝑟) of 𝐺×𝑅 into 𝐺 such that (i) (𝑔+ℎ)𝑟=𝑔𝑟+ℎ𝑟 and (ii) 𝑔(𝑟𝑠)=(𝑔𝑟)𝑠 for all 𝑔,ℎ∈𝐺 and 𝑟,𝑠∈𝑅. A subgroup (normal subgroup) 𝐻 of a right 𝑅-group 𝐺 is called an 𝑅-subgroup (ideal) of 𝐺 if ℎ𝑟∈𝐻 for all ℎ∈𝐻 and 𝑟∈𝑅.

Let 𝐺 be a right 𝑅-group. An element 𝑔∈𝐺 is called a generator of 𝐺 if 𝑔𝑅=𝐺 and 𝑔(𝑟+𝑠)=𝑔𝑟+𝑔𝑠 for all 𝑟,𝑠∈𝑅. 𝐺 is said to be monogenic if 𝐺 has a generator.

𝐺 is said to be simple if 𝐺≠{0}, and 𝐺 and {0} are the only ideals of 𝐺.

A monogenic right 𝑅-group G is said to be a right 𝑅-group of type 0 if 𝐺 is simple.

The annihilator of a right 𝑅-group 𝐺, denoted by (0∶𝐺), is defined as (0∶𝐺)={𝑎∈𝑅∣𝐺𝑎={0}}.

Lemma 2.1. The constant part of 𝑅 is right quasiregular.

Lemma 2.2. A nilpotent element of 𝑅 is right quasiregular.

Theorem 2.3. 𝐽^𝑟_1/2(𝑅) is the largest right quasiregular right ideal of 𝑅.

Theorem 2.4. 𝐽^𝑟₀(𝑅) is the largest right quasiregular ideal of 𝑅.

Theorem 2.5. 𝐽^𝑟₀(𝑅) is the intersection of all right 0-primitive ideals of 𝑅.

Theorem 2.6. Let 𝑃 be an ideal of 𝑅. 𝑃 is a right 0-primitive ideal of 𝑅 if and only if 𝑅/𝑃 is a right 0-primitive near-ring.

Proposition 2.7. Let 𝐺 be a right 𝑅-group of type 0 and 𝑔₀ a generator of 𝐺. Then (0∶𝑔₀)∶={𝑟∈𝑅∣𝑔₀𝑟=0} is a right 0-modular right ideal of 𝑅.

Proposition 2.8. Let 𝐺 be a right 𝑅-group. 𝐺 is a right 𝑅-group of type 0 if and only if there is a maximal right modular right ideal 𝐾 of 𝑅 such that 𝐺 is 𝑅-isomorphic to 𝑅/𝐾.

Proposition 2.9. Let 𝑃 be an ideal of a zero-symmetric near-ring 𝑅. 𝑃 is right 0-primitive if and only if 𝑃 is the largest ideal of 𝑅 contained in (0∶𝐺) for some right 𝑅-group 𝐺 of type 0.

Let 𝑄 be a mapping which assigns to each near-ring 𝑅 an ideal 𝑄(𝑅) of 𝑅. Such mappings are called ideal-mappings. We consider the following properties which 𝑄 may satisfy:

(H1) ℎ(𝑄(𝑅))⊆𝑄(ℎ(𝑅)) for all homomorphisms ℎ of 𝑅;

(H2) 𝑄(𝑅/𝑄(𝑅))={0} for all 𝑅;

𝑄 is r-hereditary if 𝐼∩𝑄(𝑅)⊆𝑄(𝐼) for all ideals 𝐼 of 𝑅;

𝑄 is s-hereditary if 𝑄(𝐼)⊆𝐼∩𝑄(𝑅) for all ideals 𝐼 of 𝑅;

𝑄 is ideal-hereditary if it is both 𝑟-hereditary and 𝑠-hereditary, that is, if 𝑄(𝐼)=𝐼∩𝑄(𝑅) for all ideals 𝐼 of 𝑅;

𝑄 is idempotent if 𝑄(𝑄(𝑅))=𝑄(𝑅) for all 𝑅;

𝑄 is complete if 𝑄(𝐼)=𝐼 and 𝐼 is an ideal of 𝑅 that implies 𝐼⊆𝑄(𝑅).

With 𝑄 we associate two classes of near-rings ℝ_𝑄 and 𝕊_𝑄 defined by ℝ_𝑄∶={𝑅∣𝑄(𝑅)=𝑅}, 𝕊_𝑄∶={𝑅∣𝑄(𝑅)={0}}, and are called a 𝑄-radical class and a 𝑄-semisimple class, respectively.

An ideal-mapping 𝑄 is a Hoehnke radical (𝐻-radical) if it satisfies conditions (H1) and (H2).

An ideal-mapping 𝑄 is a Kurosh-Amitsur radical (KA-radical) if it is a complete idempotent 𝐻-radical.

Let 𝕄 be a class of near-rings. Classes of near-rings are always assumed to be abstract, that is, they contain the one element near-ring and are closed under isomorphic copies. With every near-ring 𝑅, we associate two ideals of 𝑅, depending on 𝕄. These ideals are defined by the following:𝕄(𝑅)∶=Σ{𝐼∣𝐼isanidealof𝑅,𝐼∈𝕄},(𝑅)𝕄∶=∩{𝐼∣𝐼isanidealof𝑅,𝑅/𝐼∈𝕄}. (2.1) The mapping 𝑃 defined by 𝑃(𝑅)∶=(𝑅)𝕄 is always an 𝐻-radical and is called the 𝐻-radical corresponding to 𝕄.

From Theorems 2.5 and 2.6, we have the following.

Proposition 2.10. 𝐽^𝑟₀ is an 𝐻-radical corresponding to the class of all right 0-primitive near-rings.

3. Properties of the Radical 𝐽^𝑟₀

If (𝐴,+) is a group and 𝑇 is a subset of 𝐴, then the subgroup (normal subgroup) of 𝐴 generated by 𝑇 is denoted by ⟨𝑇⟩_𝑠(⟨𝑇⟩_𝑛).

Remark 3.1. Let 𝐺 be a right 𝑅-group. It is clear that 𝐻={𝑔∈𝐺∣𝑔𝑅={0}} is an ideal of 𝐺. So if 𝐺 is simple and 𝑔𝑅={0}, then 𝑔=0 provided 𝐺𝑅≠{0}.

Theorem 3.2. Let 𝐺 be a right 𝑅-group of type 0. Suppose that S is an invariant subnear-ring and a right ideal of 𝑅. If 𝐺𝑆≠{0}, then 𝐺 is also a right 𝑆-group of type 0.

Proof. Suppose that 𝐺𝑆≠{0}. Clearly, 𝐺 is a right 𝑆-group. Let 𝑔∈𝐺 and 𝑔𝑆∶={𝑔𝑠∣𝑠∈𝑆}⊆𝐺. Consider the normal subgroup ⟨𝑔𝑆⟩_𝑛 of (𝐺,+). Let 𝑟∈𝑅, ℎ∈⟨𝑔𝑆⟩_𝑛. Now ℎ=(𝑥₁+𝛿₁(𝑔𝑠₁)−𝑥₁)+(𝑥₂+𝛿₂(𝑔𝑠₂)−𝑥₂)+⋯+(𝑥_𝑘+𝛿_𝑘(𝑔𝑠_𝑘)−𝑥_𝑘), 𝑠_𝑖∈𝑆, 𝑥_𝑖∈𝐺, 𝛿_𝑖∈{1,−1}. Since 𝑆𝑅⊆𝑆, ℎ𝑟=(𝑥₁𝑟+𝛿₁(𝑔(𝑠₁𝑟))−𝑥₁𝑟)+(𝑥₂𝑟+𝛿₂(𝑔(𝑠₂𝑟))−𝑥₂)+⋯+(𝑥_𝑘𝑟+𝛿_𝑘(𝑔(𝑠_𝑘𝑟))−𝑥_𝑘𝑟)∈⟨𝑔𝑆⟩_𝑛. So ⟨𝑔𝑆⟩_𝑛 is an ideal of the right 𝑅-group 𝐺, and hence it is also an ideal of the right 𝑆-group 𝐺. Let 0≠ℎ∈𝐺. Suppose that ℎ𝑆={0}. Since ℎ𝑅≠{0}, ⟨ℎ𝑅⟩_𝑛 is a nonzero ideal of the right 𝑅-group 𝐺. Since 𝐺 is a simple right 𝑅-group, ⟨ℎ𝑅⟩_𝑛=𝐺. So 𝐺𝑆=⟨ℎ𝑅⟩_𝑛𝑆⊆⟨ℎ𝑆⟩_𝑛={0}, a contradiction to 𝐺𝑆≠{0}. Therefore, ℎ𝑆≠{0}. Let 𝑔₀ be a generator of the right 𝑅-group 𝐺. So 𝑔₀ is a distributive element of the right 𝑅-group 𝐺 and 𝑔₀𝑅 = 𝐺. Clearly, 𝑔₀ is a distributive element of the right 𝑆-group 𝐺 and hence 𝑔₀𝑆 is a subgroup of (𝐺,+). We have (𝑔₀𝑆)𝑅=𝑔₀(𝑆𝑅)𝑔₀𝑆. So 𝑔₀𝑆 is an 𝑅-subgroup of 𝐺. Let g ∈𝐺 and s ∈𝑆. Since 𝑔₀𝑅=𝐺, 𝑔=𝑔₀𝑟 for some 𝑟∈𝑅. So 𝑔+𝑔₀𝑠−𝑔=𝑔₀𝑟+𝑔₀𝑠−𝑔₀𝑟=𝑔₀(𝑟+𝑠−𝑟)∈𝑔₀𝑆, as 𝑆 is a normal subgroup of (𝑅,+). Therefore, 𝑔₀𝑆 is an ideal of the right 𝑅-group 𝐺 and hence 𝑔₀𝑆=𝐺. So 𝑔₀ is also a generator of the right 𝑆-group 𝐺. Let 𝐾 be a nonzero ideal of the right 𝑆-group 𝐺. Let 0≠𝑦∈𝐾. As seen above, ⟨𝑦𝑆⟩_𝑛 is a nonzero ideal of the right 𝑅-group 𝐺, and hence ⟨𝑦𝑆⟩_𝑛=𝐺. Since 𝐺=⟨𝑦𝑆⟩_𝑛⊆𝐾, 𝐺=𝐾. Therefore, {0} and 𝐺 are the only ideals of the right 𝑆-group 𝐺 and hence 𝐺 is a right 𝑆-group of type 0.

Proposition 3.3. Let 𝐺 be a right 𝑅-group of type 0 and let T be a right quasiregular invariant subnear-ring of 𝑅. If T is a right ideal of 𝑅, then 𝐺𝑇={0}.

Proof. Suppose that 𝑇 is a right ideal of 𝑅 and 𝑔₀ is a generator of 𝐺. So 𝑔₀(𝑟+𝑠)=𝑔₀𝑟+𝑔₀𝑠 for all 𝑟,𝑠∈𝑅 and 𝑔₀𝑅=𝐺. Now 𝐿∶=(0∶𝑔₀)={𝑟∈𝑅∣𝑔₀𝑟=0} is a right 0-modular right ideal of 𝑅. Therefore, 𝐿 contains the largest right quasiregular right ideal of 𝑅. Since 𝑇 is a right quasiregular right ideal of 𝑅, 𝑇⊆𝐿, that is, 𝑔₀𝑇={0}. Let 𝑔∈𝐺 and 𝑡∈𝑇. Now 𝑔=𝑔₀𝑟 for some 𝑟∈𝑅. 𝑔𝑡=𝑔₀(𝑟𝑡)=0, as 𝑟𝑡∈𝑇. Therefore, 𝐺𝑇={0}.

Since 𝑅_𝑐 is right quasiregular in 𝑅, we have the following.

Corollary 3.4. If 𝑅_𝑐 is a normal subgroup of (𝑅,+), then 𝐺𝑅_𝑐={0} for all right 𝑅-groups 𝐺 of type 0.

Corollary 3.5. Let 𝑅∈ℱ. If 𝐺 is a right 𝑅-group of type 0, then 𝐺𝐽^𝑟₀(𝑅)={0}.

Proof. Let 𝐺 be a right 𝑅-group of type 0. We have that 𝐼∶=𝐽^𝑟₀(𝑅) is the largest right quasiregular ideal of 𝑅. Since 𝑅_𝑐 is a right quasiregular ideal of 𝑅,𝑅_𝑐⊆𝐼. So 𝐼 is an invariant ideal of 𝑅. Therefore, by Proposition 3.3, 𝐺𝐼={0}.

Proposition 3.6. Let 𝑅∈ℱ. Let 𝐼 be an ideal of 𝑅 and 𝐾∶=𝐼+𝑅_𝑐. If 𝐺 is a right 𝐾-group of type 0, then 𝐺 is a right 𝐼-group of type 0.

Proof. Suppose that 𝐺 is a right 𝐾-group of type 0 and 𝑔₀ is a generator of 𝐺. So 𝑔₀ is distributive over 𝐾 and 𝑔₀𝐾=𝐺. Let 𝐾_𝑐 be the constant part of 𝐾. Since 𝐾_𝑐=𝑅_𝑐 is a normal subgroup of 𝐾, by Corollary 3.4, 𝐺𝑅_𝑐={0}. Clearly, 𝐺 is a right 𝐼-group. Now 𝐺=𝑔₀𝐾=𝑔₀(𝐼+𝑅_𝑐)=𝑔₀𝐼, and hence 𝑔₀ is a generator of the right 𝐼-group 𝐺. Let 𝐻 be a nonzero ideal of the right 𝐼-group 𝐺. Let ℎ∈𝐻 and 𝑘∈𝐾. 𝑘=𝑖+𝑟_𝑐, 𝑖∈𝐼, 𝑟_𝑐∈𝑅_𝑐 and ℎ=𝑔₀𝑡, 𝑡∈𝐼. ℎ𝑘=𝑔₀𝑡(𝑖+𝑟_𝑐)=𝑔₀((𝑡(𝑖+𝑟_𝑐)−𝑡𝑖)+𝑡𝑖)=𝑔₀(𝑡(𝑖+𝑟_𝑐)−𝑡𝑖)+𝑔₀(𝑡𝑖)=0+(𝑔₀𝑡)𝑖=ℎ𝑖∈𝐻. Therefore, 𝐻 is a nonzero ideal of the right 𝐾-group 𝐺 and hence 𝐻=𝐺. So 𝐺 is a right 𝐼-group of type 0.

We show now that the Hoehnke radical 𝐽^𝑟₀ is complete in the variety ℱ.

Theorem 3.7. Let 𝑅∈ℱ. If 𝐼 is an ideal of 𝑅 and 𝐽^𝑟₀(𝐼)=𝐼, then 𝐼⊆𝐽^𝑟₀(𝑅).

Proof. Let 𝐼 be an ideal of 𝑅 and 𝐽^𝑟₀(𝐼)=𝐼. Suppose that 𝐼⊆𝐽̸^𝑟₀(𝑅). So 𝐾∶=𝐼+𝑅_𝑐 is an ideal of 𝑅 and 𝐾⊆𝐽̸^𝑟₀(𝑅). We get a right 𝑅-group 𝐺 of type 0 such that 𝐺𝐾≠{0}. Since 𝐾 is an invariant ideal of 𝑅, by Theorem 3.2, 𝐺 is a right 𝐾-group of type 0. Therefore, by Proposition 3.6, 𝐺 is a right 𝐼-group of type 0. This is a contradiction to the fact that 𝐽^𝑟₀(𝐼)=𝐼. Therefore, 𝐼⊆𝐽^𝑟₀(𝑅).

Theorem 3.8. 𝐽^𝑟₀ is a complete Hoehnke radical in the variety ℱ.

Theorem 3.9. 𝐽^𝑟₀ is a complete Hoehnke radical in the class of all zero-symmetric near-rings.

Theorem 3.10. Suppose that 𝑆 is an invariant subnear-ring of 𝑅. If 𝐺 is a right 𝑆-group of type 0, then 𝐺 is also a right 𝑅-group of type 0.

Proof. Suppose that 𝐺 is a right 𝑆-group of type 0 and 𝑔₀ is a generator. We have that 𝑔₀ is distributive over 𝑆 and 𝑔₀𝑆=𝐺. For 𝑔∈𝐺 and 𝑟∈𝑅, define 𝑔𝑟∶=𝑔₀(𝑠𝑟), if 𝑔=𝑔₀𝑠, 𝑠∈𝑆. We show now that this operation is well defined. Suppose that 𝑔=𝑔₀𝑠=𝑔₀𝑡, 𝑠,𝑡∈𝑆. Let 𝑟∈𝑅 and ℎ∶=𝑔₀(𝑠𝑟)−𝑔₀(𝑡𝑟). Now ℎ𝑘=(𝑔₀(𝑠𝑟)−𝑔₀(𝑡𝑟))𝑘=𝑔₀((𝑠𝑟)𝑘)−𝑔₀((𝑡𝑟)𝑘)=𝑔₀(𝑠(𝑟𝑘))−𝑔₀(𝑡(𝑟𝑘))=𝑔(𝑟𝑘)−𝑔(𝑟𝑘)=0 for all 𝑘∈𝑆. Therefore, ℎ𝑆={0}, and hence ℎ=0, that is, 𝑔₀(𝑠𝑟)=𝑔₀(𝑡𝑟). We show that 𝐺 is a right 𝑅-group of type 0. It is clear that 𝐺 is a right 𝑅-group. 𝑔₀=𝑔₀𝑒 for some 𝑒∈𝑆. Now 𝐺⊇𝑔₀𝑅=𝑔₀(𝑒𝑅)⊇𝑔₀(𝑒𝑆)=𝑔₀𝑆=𝐺. So 𝑔₀𝑅=𝐺. Let 𝑝,𝑞∈𝑅 and 𝑥=𝑔₀(𝑝+𝑞)−(𝑔₀𝑝+𝑔₀𝑞). 𝑥𝑠=(𝑔₀(𝑝+𝑞)−(𝑔₀𝑝+𝑔₀𝑞))𝑠=(𝑔₀(𝑝+𝑞))𝑠−((𝑔₀𝑝+𝑔₀𝑞))𝑠=𝑔₀(𝑝𝑠+𝑞𝑠)−(𝑔₀𝑝𝑠+𝑔₀𝑞𝑠)=(𝑔₀(𝑝𝑠)+𝑔₀(𝑞𝑠))−(𝑔₀(𝑝𝑠)+𝑔₀(𝑞𝑠))=0 for all 𝑠∈𝑆. Therefore, 𝑥=0, and hence 𝑔₀ is a generator of the right 𝑅-group 𝐺. It can be easily verified that the action of 𝑅 on 𝐺 is an extension of the action of 𝑆 on 𝐺. So an ideal of the right 𝑅-group 𝐺 is also an ideal of the right 𝑆-group 𝐺. Since the right 𝑆-group 𝐺 has no nontrivial ideals, the right 𝑅-group 𝐺 also has no nontrivial ideals. Therefore, 𝐺 is also a right 𝑅-group of type 0.

We show now that the Hoehnke radical 𝐽^𝑟₀ is idempotent in the variety ℱ.

Theorem 3.11. Let 𝑅∈ℱ. Then 𝐽^𝑟₀(𝐽^𝑟₀(𝑅))=𝐽^𝑟₀(𝑅).

Proof. Let 𝐼∶=𝐽^𝑟₀(𝑅). 𝐼 is the largest right quasiregular ideal of 𝑅. Since 𝑅_𝑐 is a right quasiregular ideal of 𝑅, 𝑅_𝑐⊆𝐼. So 𝐼 is an invariant ideal of 𝑅. Suppose that 𝐽^𝑟₀(𝐼)≠𝐼. So there is a right 𝐼-group 𝐺 of type 0. By Theorem 3.10, 𝐺 is an 𝑅-group of type 0. Now, by Corollary 3.5, 𝐺𝐼=𝐺𝐽^𝑟₀(𝑅)={0}. This is a contradiction to the fact that 𝐺 is an 𝐼-group of type 0. Therefore, 𝐽^𝑟₀(𝐼)=𝐼, that is, 𝐽^𝑟₀(𝐽^𝑟₀(𝑅))=𝐽^𝑟₀(𝑅).

From Theorems 3.7 and 3.11, we have the following.

Theorem 3.12. 𝐽^𝑟₀ is a Kurosh-Amitsur radical in the variety ℱ.

Theorem 3.13. 𝐽^𝑟₀ is a Kurosh-Amitsur radical in the class of all zero-symmetric near-rings.

Theorem 3.14. 𝐽^𝑟₀ is not s-hereditary in the class of all zero-symmetric near-rings.

Proof. Consider 𝐺∶=𝑍₈, the group of integers under addition modulo 8. Now 𝑇: 𝐺→𝐺, defined by 𝑇(𝑔)=5𝑔, for all 𝑔∈𝐺, is an automorphism of 𝐺. 𝑇 fixes 0, 2, 4, and 6, and maps 1 to 5 and 3 to 7. 𝐴∶={𝐼,𝑇} is an automorphism group of 𝐺. {0}, {2}, {4}, {6}, {1,5}, and {3,7} are the orbits. Let 𝑅 be the centralizer near-ring 𝑀_𝐴(𝐺), the near-ring of all self maps of 𝐺 which fix 0 and commute with 𝑇. An element of 𝑅 is completely determined by its action on {1,2,3,4,6}. An element 𝑓∈𝑅 maps 2𝐺 into 2𝐺 and 𝑓(1) and 𝑓(3) are arbitrary in 𝐺. This example was considered in [9] and showed that 𝑃∶=(0∶2𝐺)={𝑓∈𝑅∣𝑓(ℎ)=0,forallℎ∈2𝐺} is the only nontrivial ideal of 𝑅. Let 𝑓₀ be the element of 𝑃 which fixes all the elements in 𝐺−2𝐺. Clearly 𝑓−𝑓₀𝑓∈(2𝐺∶𝐺)={𝑡∈𝑅∣𝑡(𝐺)⊆2𝐺} for all 𝑓∈𝑅. Since (2𝐺∶𝐺) is a proper right ideal of 𝑅, 𝑓₀ is not right quasiregular in 𝑅. So 𝑃 is not a right quasiregular ideal of 𝑅. Since 𝑅 is a near-ring with identity, it is not right quasiregular. Therefore, {0} is the largest right quasiregular ideal of 𝑅, and hence 𝐽^𝑟₀(𝑅)={0}. So 𝑅 is 𝐽^𝑟₀-semisimple. It is shown in [9] that 𝐾∶=(4𝐺∶𝐺)_𝑃 is a nonzero ideal of 𝑃 and 𝐾²={0}. Since a nil ideal is right quasiregular, 𝐾 is a right quasiregular ideal of 𝑃. Therefore, {0}≠𝐾⊆𝐽^𝑟₀(𝑃) and hence 𝑃 is not 𝐽^𝑟₀-semisimple. So 𝐽^𝑟₀ is not 𝑠-hereditary in the class of all zero-symmetric near-rings.

Corollary 3.15. 𝐽^𝑟₀ is not 𝑠-hereditary in the class of all near-rings.

Theorem 3.16. 𝐽^𝑟₀ is not an ideal-hereditary radical in the class of all zero-symmetric near-rings.

It is not known to the authors whether 𝐽^𝑟₀ is a KA-radical in the class of all near-rings. 𝐽^𝑟₀ may fail to be idempotent and thus Kurosh-Amitsur in the class of all near-rings.