EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6576 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Kurosh-Amitsur Completely Prime Radical for Near-rings Kilaru J. Lakshminarayana1,∗, V.B.V.N. Prasad1, Srinivasa Rao Ravi2, Siva Prasad Korrapati2 A.V. Ramakrishna3 1 Department of Engineering Mathematics Koneru Lakshmaiah Education Foundation, Vaddeswaram-522502, Guntur (Dist.), Andhra Pradesh, India 2 Department of Mathematics University College of Sciences, Acharya Nagarjuna University, Nagarjuna Nagar-522510 Guntur (Dist.), Andhra Pradesh, India 3 Department of Mathematics R.V.R and J.C College of Engineering,Chowdavaram-522019, Guntur (Dist.),Andhra Pradesh, India Abstract. Two generalizations of the completely prime radical of rings to near-rings, namely the completely prime radical of near-rings and the completely equiprime radical of near-rings were introduced and studied. First one is not a Kurosh-Amitsur radical but the second one is a special radical in near-rings. In this article another generalization of the completely prime radical of rings is introduced in near- rings using right modules of near-rings. For this completely prime right N -groups of type-r(1) are introduced in near-rings, N is a near-ring. Making use of these right N -groups of type-r(1), the completely prime radical of near-rings of type-r(1) is introduced. It is observed that the completely prime radical of type-r(1) is a Kurosh-Amitsur radical. 2020 Mathematics Subject Classifications: 16Y30 Key Words and Phrases: Near-ring, right N -group, rightN -groups of type r(1), completely Prime radical of type r(1) 1. Introduction In this article, we consider right zero-symmetric near-rings. The concept of the com- pletely prime radical, originally developed for rings, was extended to near-rings by N. J. Groenewald [1]. He demonstrated that, analogous to the prime radical in near-rings, the completely prime radical of near-rings does not satisfy the properties of a Kurosh-Amitsur radical. Based on the notion of equiprime ideals in near-rings, a further generalization of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6576 Email addresses: 2002511005@kluniversity.in (Kilaru J. Lakshminarayana), vbvnprasad@kluniversity.in (V.B.V.N. Prasad), dr rsrao@yahoo.com (S. Rao Ravi), siva235prasad@yahoo.co.in (S. P. Korrapati), amathi7@gmail.com (A. V. Ramakrishna) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) K. J. Lakshminarayana et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6576 2 of 7 completely prime radical of rings called the completely equiprime radical was introduced in [2]. It was shown that this radical forms a special class of radicals within the framework of near-rings. Completely prime modules of rings were introduced in [3]. In [4], completely prime modules of rings were extended to (left) N -groups, N is a near-ring and the corresponding radical is the completely prime radical of near-ring which is not a Kurosh-Amitsur radical of near-rings. Right module theoretic characterization of radicals of near-rings was studied in [5]. Prime right N -groups were introduced and their correspoding radicals were studied in [6], [7] and [8]. In this article, the concept of completely prime module is extended to (right) N -groups which leads to another completely prime radical of near-rings and is a Kurosh-Amitsur radical of near-rings. A group (T,+) is a right N -group if there is a mapping (t, a) → ta of T ×N into T such that: (i) t(ab) = (ta)b; (ii) t(a+ b) = ta+ tb for all t ∈ T, a, b ∈ N . I is a right N -group for any right ideal I of N under the multiplication in N . Moreover, for a right ideal I of N , N/I is a right N -group under (a+ I)b = ab+ I, a, b ∈ N . A subgroup (normal subgroup) D of the right N -group T is a right N -subgroup (ideal) of T if da ∈ D for all d ∈ D, a ∈ N . t ∈ T is called a distributive element of the right N -group T if t(a + b) = ta + tb for all a, b ∈ N . 2. Completely Prime Right N-groups of type r(1) Unless or otherwise specified near-rings considered are zero-symmetric right near-rings. Definition 1. A right N -group G with GN ̸= {0} and G0 = {0} is called a completely prime N -group of type r(1) if (i) every non-zero right N -subgroup of G contains a distributive element g0(̸= 0) and; (ii) gr = 0 implies either g = 0 or Gr = {0} for all g ∈ G, r ∈ R. Remark 1. Let R be a ring and M be a completely prime (right) R-module. Then M is also a completely prime right R-group of type r(1), when R is considered as a near-ring. Example 1. Let N be a near-field. It is clear that N is a completely prime right N -group of type r(1). The following Proposition is obvious in view of the above definition. K. J. Lakshminarayana et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6576 3 of 7 Proposition 1. Let G be right N -group with GN ̸= {0} and each non-zero N -subgroup of G has a non-zero distributive element. Then G is a completely prime N -group of type r(1) if and only if (g : 0) = (G : 0) for all 0 ̸= g ∈ G. Proof. Suppose that G completely prime right N -group of type r(1). Let 0 ̸= g ∈ G and r ∈ N and gr = 0. By definition of completely prime right N -group of type r(1), Gr = {0}. So (g : 0) ⊆ (G : 0). Since (G : 0) ⊆ (g : 0), we have (g : 0) = (G : 0). Conversely suppose that (g : 0) = (G : 0) for all 0 ̸= g ∈ G. Let 0 ̸= g ∈ G, r ∈ N and gr = 0. By assumption, Gr = {0}. So G completely prime right N -group of type r(1). Proposition 2. Let G be a completely prime right N -group of type r(1) and H be N - subgroup of G. Then H is a completely prime right N -group of type r(1). Proof. This follows from the definition of completely prime right N -group. Proposition 3. Let G be a completely prime right N -group of type r(1). If N is a ring, then G is a completely prime ring N -module. Proof. Obvious from the definition of the Completely prime right N -group. Proposition 4. Let G be a completely prime right N -group of type r(1) and I be an ideal of N and GI = {0}.Then G has a right N -subgroup H which is a Completely prime N/I-group of type r(1). Proof. Let g0 be a distributive element of the right N -group G. Consider g0N := {g0a | a ∈ N}. For g1, g2 ∈ g0N , g1 = g0b and g2 = g0c for some b, c ∈ N . Now g1 − g2 = g0(b− c) ∈ g0N . So g0N is a subgroup of (G,+). Since (g0N)N ⊆ g0(NN) ⊆ g0N , g0N is a right N -subgroup of G. We claim that g0N is a right N/I-group, under g(x+ I) = gx, g ∈ g0N and x+ I ∈ N/I. Let g := g0a ∈ g0N and x+ I, y+ I ∈ N/I and x+ I = y+ I. We have x − y ∈ I. gx = (g0a)x = g0(ax) = g0(a((x − y) + y)) − g0(ay) + g0(ay) = g0(a((x − y) + y) − ay) + g0(ay) = 0 + (g0a)y) = gy. So the operation is well defined and g0N is a right N/I-group. It is clear that g ′ ∈ g0N is a distributive element of the right N/I-group g0N if and only if g ′ is a distributive element of the right N -group G. Since every right N/I-subgroup of g0N is an N subgroup of G, every non-zero N/I- subgroup of g0N contains a non-zero distributive element. Let 0 ̸= g3 ∈ g0N . We have (g3 : 0)N/I = {n + I ∈ N/I | g3(n + I) = 0} = {n + I ∈ N/I | g3n = 0} = (g3 : 0)N/I = (g0N : 0)N/I = (g0N : 0)N/I . Hence g0N is a completely prime right N/I-group of type r(1). Proposition 5. Let N be a near-ring and I is an ideal of N and G be completely prime right N/I-group type r(1). Then G is a completely prime right N -group of type r(1). Proof. N is a near-ring and I is an ideal of N and G is a completely prime right N/I-group type r(1). Define gx := g(x + I) for all g ∈ G, x ∈ N . This makes G a right N -group. Let H be a non-zero right N -subgroup of G. It is clear that H is also a non-zero K. J. Lakshminarayana et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6576 4 of 7 right N/I-subgroup of G and it contains a non-zero distributive element h0. We have h0(x + y) = h0(x + y + I) = h0((x + I) + (y + I)) = h0(x + I) + h0(y + I) = h0x + h0y. So h0 is also a non-zero distributive element of the right N - subgroup H of G. Let 0 ̸= g1, 0 ̸= g2 ∈ G. Now (g1 : 0)N = {x ∈ N | g1x = 0} = {x ∈ N | g1(x+ I) = 0} = {x ∈ N | g2(x+ I) = 0} = {x ∈ N | g2x = 0} = (g2 : 0)N . Hence G is a completely prime right N -group of typer(1). Proposition 6. Let G be a Completely prime right N -group of type 1 and (G : 0) = {x ∈ N | gx = 0 for all g ∈ G} be the annihilator of G. Then there is a largest ideal of N contained in (G : 0). Proof. We have 0 ∈ (G : 0). Let I, J be ideals of N contained in (G : 0). By definition, G has a non zero distributive element g0. We have (g0 : 0) = (G : 0) and g0(y + z) = g0y + g0z = 0 + 0 = 0 for all y ∈ I and z ∈ J . So g0(I + J) = 0. Therefore I + J ⊆ (g0 : 0) = (G : 0). Hence there is a largest ideal of N contained in (G : 0). (G : 0) denotes the largest ideal of N contained in (G : 0). Definition 2. Let G be a completely prime right N -group of type 1 and P := (G : 0). Then P is called a completely prime ideal of N of type r(1). Definition 3. A near-ring N is called a completely prime near-ring of type r(1) if it’s zero ideal is a completely prime ideal of type r(1). Corollary 1. Let N be a near-ring and P be a completely prime ideal of N of type r(1). Then N/P completely prime near-ring of type r(1). Proof. Since P is a completely prime ideal of N of type r(1), there is a completely prime right N -group G of type r(1) and P is the largest ideal of N contained in (G : 0). By Propostion 4, there is a right N -subgroup H of G which is a completely prime N/P group of type r(1), where h(x + P ) := hx for all h ∈ H, x ∈ N and (H : 0)N/P = (G : 0)N/P . Therefore the zero ideal, (0), is the largest ideal of N/P contained in (H : 0)N/P . So (0) is a completely prime ideal of N/P of type r(1). Hence N/P is a completely prime near-ring of type r(1). Corollary 2. Let N be a near-ring and P be an ideal of N and N/P be a completely prime near ring of type r(1). Then P is a completely prime ideal of N of type r(1). Proof. P is an ideal of a near-ring N and N/P is a completely prime near-ring of type r(1). So the zero ideal, (0), is a completely prime ideal of N/P of type r(1).Therefore there is a completely prime right N/P - group G of type r(1) such that the zero ideal of N/P is the largest ideal of N/P contained in (G : 0)N/P . By Propostion 5, G is also a completely prime right N -group of type r(1), where gx := g(x + I) for all g ∈ G and x ∈ N and (G : 0)N/P = (G : 0)N/P . Therefore P is the largest ideal of N contained in (G : 0)N . Hence P is a completely prime ideal of N of type r(1). K. J. Lakshminarayana et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6576 5 of 7 Definition 4. Let N be nearring. Define Pcr(1)(N) := ∩{I | I is a completely prime ideal of N of type r(1)} and Pcr(1)(N) := N if N has no completely prime ideals of type r(1). Pcr(1) is the completely prime radical of type r(1). Proposition 7. Pcr(1) is a H-radical corresponding to the class of all completely prime zero-symmetric near rings of type r(1). Proof. Let Mcr(1) := {N | N is completely prime zero-symmetric near rings of type r(1)}. For a near-ring N , we have (N)Mcr(1) := ∩{J | J is an ideal of N and N/J ∈ Mcr(1)}. Now Q defined by Q(N) := (N)Mcr(1), is the H-radical corresponding to the class of all completely prime zero-symmetric near rings of type r(1). From the Corollaries 1 and 2, Pcr(1) = Q. Hence Pcr(1) is the H radical corresponding to the class of all completely prime zero-symmetric near rings of type r(1). Lemma 1. Let G be a Completely prime right N -group of type (1) and I be an ideal of N and GI ̸= {0}. Then G is a Completely prime right I-group of type 1 and (G : 0)I ⊇ (G : 0)N ∩ I. Proof. Since G is a right N -group and GI ̸= {0}, G is also a non-trivial I-group under restriction. Let H be a non-zero right I-subgroup of G and 0 ̸= h ∈ H. Now hI ̸= {0} as GI ̸= {0}. Let T be the subgroup of G generated by hI := {ha | a ∈ I}. T is a right I-subgroup of G as well as right N -subgroup of G. Being right N -subgroup of G, T has a distributive element t0. So t0(x+ y) = t0x+ t0y for all x, y ∈ N . Hence t0 is a distributive element of the right I-group T ⊆ H. For 0 ̸= g1, 0 ̸= g2 ∈ G, (g1 : 0)I = (g1 : 0)N ∩ I = (g2 : 0)N ∩ I = (g2 : 0)I . Also (G : 0)I = (g1 : 0)I = (g1 : 0)N ∩ I = (G : 0)N ∩ I. Therefore (G : 0)I ⊇ (G : 0)N ∩ I. Lemma 2. Let G be a Completely prime right I-group of type 1 and I, an ideal of a near-ring N . Then H := g0I is a completely prime right N -group of type r(1) and (G : 0)I = (H : 0)I ⊇ (H : 0)N ∩ I, g0 is a distributive element of the right I-group G. Proof. It is clear that H = g0I := {g0a | a ∈ I} is a subgroup of (G,+) and hence H is a right I-subgroup of G. For g0a ∈ H define (g0a)x := g0(ax) for all x ∈ N . We claim that this operation is well defined. Let g0a = g0b, a, b ∈ I and x ∈ N and c ∈ I. [g0(ax)−g0(bx)]c = g0(ax)c−g0(bx)c = (g0a)(xc)−(g0b)(xc) = (g0a−g0b)(xc) = 0(xc) = 0. If g0(ax) − g0(bx) ̸= 0 then GI = 0, a contraction. So the operation is well defined and H is a right N -group with HN ̸= (0). Let △ be a non zero right N -subgroup of H. It is clear that △ is also a right I-subgroup of H and hence there is a h0 ∈ △ such that h0(a + b) = h0a + h0b for all a, b ∈ I. Now h0 := g0d for some d ∈ I. Let x, y ∈ N . [h0(x + y) − (h0x + h0y)]c = (g0d)(x + y)c − ((g0d)xc + (g0d)yc) = (g0d)(xc + yc) − ((g0d)(xc) + (g0d)(yc)) = h0(xc) + h0(yc)− (h0(xc) + h0(yc)) = 0 for all c ∈ I. Therefore h0(x + y) = h0x + h0y and hence a distributive element of the right N -subgroup △. Let 0 ̸= g0a, 0 ̸= g0b ∈ H, a, b ∈ I. (g0a : 0)N = (g0a : 0)I ∩N = (g0b : 0)I ∩N = (g0b : 0)N . Therefore H is a completely prime N -group of type r(1). We have (H : 0)I = (H : 0)N ∩I. So (G : 0)I = (H : 0)I ⊇ (H : 0)N ∩ I. K. J. Lakshminarayana et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6576 6 of 7 Theorem 1. Pcr(1) is a complete H-radical. Proof. Suppose that Pcr(1)(J) = J , J is an ideal of N . We claim that J ⊆ Pcr(1)(N). On the contrary suppose that J ̸⊆ Pcr(1)(N). So there is completely prime right N -group G of type r(1) such that GJ ̸= {0}. By Lemma 1, G is a completely prime right J-group of type r(1). This is a contradiction to Pcr(1)(J) = J . Therefore J ⊆ Pcr(1)(N). Hence Pcr(1) is a complete H-radical. Theorem 2. The H-radical Pcr(1) is r-hereditary. Proof. If J is an ideal of a near-ring N and G is a Completely prime right J-group of type 1 then by Lemma 2, there is a right J-subgroup H of G such that H is a completely prime right N -group of type r(1) and (G : 0)J = (H : 0)J ⊇ (H : 0)N ∩ J . Therefore Pcr(1)(J) ⊇ Pcr(1)(N) ∩ J . Hence Pcr(1) is r-hereditary. Theorem 3. Pcr(1) is an idempotent H-radical. Proof. Let N be a near-ring and J := Pcr(1)(N). From Theorem 2, Pcr(1)(J) ⊇ Pcr(1)(N) ∩ J . Taking Pcr(1)(N) for J in the above inclusion, we have Pcr(1)(Pcr(1)(N)) = Pcr(1)(N). Therefore Pcr(1) is an idempotent H-radical. From Proposition 7 and Theorems 3 and 1, we have the following: Theorem 4. The H-radical Pcr(1) is a Kurosh-Amitsur radical in the class of all zero- symmetric near-rings. Proof. Since an idempotent, complete H-radical is a Kurosh-Amitsur radical, from Proposition 7 and Theorems 1 and 3, Pcr(1) is a Kurosh-Amitsur radical. 3. Conclusions Some important Kurosh-Amitsur prime radicals of rings fails to be Kurosh-Amitsur radicals, when they are generalized to near-rings using left N-groups. To see whether the situation will be different with right N -groups was studied in [6], [5] and [8]. As proved in [6], right N -groups play significant role in generalizing the prime radicals of rings to near-rings. This fact was completely established in [5] in general. In [8], the prime radical of rings was generalized to near-rings using right N -groups which is also a Kurosh-Amitsur radical of near-rings. In this article, the completely prime radical of rings which is a Kurosh-Amitsur radical of rings is generalized to near-rings using right N -groups which is also a Kurosh-Amitsur radical of near-rings. K. J. Lakshminarayana et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6576 7 of 7 References [1] N. J. Groenewald. Note on the completely prime radical in near-rings. Near-rings and Near-fields, North-Holland Mathematics Studies, 137:97–100, 1987. [2] G. L. Booth and N. J. Groenewald. Equiprime left ideals and equiprime n-groups of a near-ring. Contributions to General Algebra, 8:25–38, 1992. [3] N. J. Groenewald. Completely prime submodules. 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