Prime ideals and ideal symmetry in near- rings

The most interesting and active field of current research in mathematics is theory of near- ring due to its wide application in coding theory, group theory, geometry, cryptography and block- designing etc. Near rings are one of the generalized structure of rings. Modern ring theory has very active mathematical discipline and studying rings in their own right. In this paper we study about 3- prime ideals and radicals in a Near-ring. Also, we study about 3- prime ideal and ideal symmetry in a Near-ring and its inter relationship between them.

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Introduction

The most interesting and active field of current research in mathematics is theory of near-ring due to its wide application in coding theory, group theory, geometry, cryptography and blockdesigning etc. Near rings are one of the generalized structure of rings, The study and research on near-rings is very systematic and continuous. Near-rings around in all directions of mathematics and continuous research is being conducted, which show that their structure has power and beauty all its own. Modern ring theory has very active mathematical discipline and studying rings in their own right. The key ideas important to near-ring were formed by L.E. Dickson in 1905. Actually he gave the concept of near-field. In 1930 Wieland studied nearrings which were not near-fields. In this paper we study about 3-prime ideals and radicals in a Near-ring. Also, we study about 3-prime ideal and ideal symmetry in a Near-ring and its inter relationship between them.

𝟏prime ideal

An ideal 𝐼 of a near-ring 𝑁 is called 3prime ideal, if 𝑎, 𝑏 ∈ 𝑁 and 𝑎𝑁𝑏 ⊆ 𝐼 implies either 𝑎 ∈ 𝐼 𝑜𝑟 𝑏 ∈ 𝐼 and 𝑁 is called 3prime near ring if {0} is a 3prime ideal of 𝑁.

Example: Let 𝑁 = {𝑜, 𝑎, 𝑏, 𝑐} be a near-ring with addition and multiplication are defined as: 0 + 0 = 0,0 + 𝑎 = 𝑎, 0 + 𝑏 = 𝑏, 0 + 𝑐 = 𝑐. 𝑎 + 0 = 𝑎, 𝑎 + 𝑎 = 0, 𝑎 + 𝑏 = 𝑐, 𝑎 + 𝑐 = 𝑏, 𝑏 + 0 = 𝑏, 𝑏 + 𝑎 = 𝑏, 𝑏 + 𝑏 = 0, 𝑏 + 𝑐 = 𝑎, 𝑐 + 0 = 𝑐, 𝑐 + 𝑎 = 𝑐, 𝑐 + 𝑏 = 𝑎 and 𝑐 + 𝑐 = 0. 0.0 = 0, 0. 𝑎 = 0, 0. 𝑏 = 0, 0. 𝑐 = 0𝑎. 0 = 𝑎, 𝑎. 𝑎 = 𝑎, 𝑎. 𝑏 = 0, 𝑎. 𝑐 = 𝑎, 𝑏. 0 = 𝑏, 𝑏. 𝑎 = 𝑏. 𝑏. 𝑏 = 𝑏, 𝑏. 𝑐 = 𝑏, 𝑐. 0 = 𝑏, 𝑐. 𝑎 = 𝑐, 𝑐. 𝑏 = 𝑏, 𝑐. 𝑐 = 𝑐. Then {𝑜, 𝑎} is a 3prime ideal of 𝑁. Proposition 1.1: Every equi-prime ideal is 3-prime ideal but every 3-prime near ring need not be an equi-prime near ring.

Note: Equi-prime ideal ⇒ 3prime ideal-⇒ prime ideal.

Example: Let (𝑁, +, . ) be any group with at least three elements. We define multiplication on N as: 𝑎. 𝑏 = 𝑎, 𝑖𝑓 𝑏 ≠ 0 𝑎𝑛𝑑 0, 𝑖𝑓 𝑏 = 0 Then (𝑁, +, . ) is a zero-symmetric near-ring. Therefore for any non zero elements 𝑥, 𝑦 ∈ 𝑁 ⇒ 𝑥𝑛𝑦 = 𝑥 ≠ 0. Hence 𝑥𝑁𝑦 ≠ 0. This shows that N is 3prime. Let 0 not element of {𝑥, 𝑦} ⊆ 𝑁 such that 𝑥 ≠ 𝑦 then for each element n of N. 𝑥𝑛𝑥 = 𝑥, if 𝑛 ≠ 0 = 𝑥𝑛𝑦 𝑎𝑛𝑑 0, if 𝑛 = 0 Thus 𝑥𝑛𝑦 = 𝑥𝑛𝑥 for all 𝑛 ∈ 𝑁 but, 𝑥 ≠ 𝑦. Hence N is not an equi-prime near-ring. Corollary 1.2: Let 𝐼 and 𝐽 be ideals of a near ring 𝑁 such that 𝐼 ⊆ 𝐽, then 𝐺 {0} (𝑁) ⊆ 𝐺 𝐼 (𝑁) ⊆ 𝐺 𝐽 (𝑁) ⊆ 𝐺 𝑁 (𝑁).

Proof: Since 𝐼 ⊆ 𝐽 so, {0} ⊆ 𝐼 ⊆ 𝐽 ⊆ 𝑁. Therefore, we have the set of edges in 𝐺 {0} (𝑁) ⊆ 𝑇he set of edges in 𝐺 𝐼 (𝑁) ⊆ The set of edges in 𝐺 𝐽 (𝑁) ⊆ 𝑇he set of edges in 𝐺 𝑁 (𝑁). Hence from these we have 𝐺 {0} (𝑁) ⊆ 𝐺 𝐼 (𝑁) ⊆ 𝐺 𝐽 (𝑁) ⊆ 𝐺 𝑁 (𝑁). Definition: Let 𝐺 be a graph of with vertex set 𝑉(𝐺). A strong vertex cut of a graph 𝐺 is a subset 𝑆 ⊆ 𝑉(𝐺) such that (𝐺 -𝑆) is totally disconnected. The strong vertex connectivity of 𝐺 is define as (𝐺) = 𝑚𝑖𝑛(𝑛 ≥ 0, where 𝐾(𝐺) = strong vertex connectivity of 𝐺 there exists a strong vertex cut 𝑆 ≤ 𝑉(𝐺) such that |𝑆| = 𝑛 ). Theorem 1.5: Let 𝐼 be an ideal of a near ring 𝑁. If 𝐼 be 3prime, then 𝐼 is a strong vertex cut of 𝐺 𝐼 (𝑁). If 𝐼 is a 3semi-prime ideal of 𝑁 and a strong vertex cut of 𝐺 𝐼 (𝑁), then 𝐼 is 3prime. Converse of this theorem does not hold if the condition that 𝐼 is 3semiprime is omitted.

Example: Consider the graph 𝐺 𝐼 (𝑉 4 ) where 𝐼 = {0}. Here {0} is a strong vertex cut of 𝐺 𝐼 (𝑍 4 ), however 𝐼 = {0} is not a 3prime ideal since 2𝑁2 ⊆ {0}𝑏𝑢𝑡 2 ∉ 𝐼 = {0}. Example: In the graph 𝐺 𝐼 (𝑁), where 𝑁 = {0, 𝑎, 𝑏, 𝑐}, 𝐼 = {0, 𝑎} we can observe that 𝐼 is a 3-prime ideal of 𝑁. Example: In the graph 𝐺 𝐼 (ℤ 6 ), 𝐼 = {0}, we observe that 𝐼 = {0} is not a 3-prime ideal of ℤ 6 . Since 𝐼 = {0} is not a strong vertex cut.

Proposition 1.6: Let 𝐼 be a minimal 3prime ideal of a near-ring 𝑁, then 𝐾(𝐺 𝐼 (𝑁)) = |𝐼|.

Proof: Let 𝐼 be a minimal 3prime ideal of a near-ring 𝑁 that is 𝐼 ≠ {0} and if there is an another 3-prime ideal 𝐽 such that {0} ⊂ 𝐽 ⊂ 𝐼 then either 𝐽 = {0} or 𝐽 = 𝐼. Since 𝐼 is a minimal 3-prime ideal so, 𝐼 is a strong vertex cut of 𝐺 𝐼 (𝑁). If we remove 𝐼 from 𝐺 𝐼 (𝑁), then it is a totally disconnected graph. Hence 𝐾(𝐺 𝐼 (𝑁)) = |𝐼|. Theorem 1.7: Let 𝐼 be a 3prime ideal of a near-ring 𝑁 and 𝑥 be a vertex in 𝐺 𝐼 (𝑁) if 𝑑𝑒𝑔(𝑥) = 𝑑𝑒𝑔(0), then 𝑥 ∈ 𝐼. Theorem 1.8: Let N be a zero-symmetric near ring 𝑁 and 𝑥 be a vertex in 𝐺 𝐼 (𝑁) if 𝑥 ∈ 𝐼, then 𝑑𝑒𝑔(𝑥) = 𝑑𝑒𝑔(0). Theorem 1.8: Let 𝑁 be a zero-symmetric near ring 𝑁 and I be a 3prime ideal of 𝑁, then 𝑥 ∈ 𝐼 if and only if 𝑑𝑒𝑔(𝑥) = 𝑑𝑒𝑔(0) in 𝐺 𝐼 (𝑁). Theorem 1.9: Let |N| =n and 𝐼 be an ideal of a near ring 𝑁 then, if 𝑁 is a zero-symmetric near ring and 𝐼 is a minimal 3prime ideal of 𝑁, then |𝐼| = 𝐾(𝐺 𝐼 (𝑁)) ≤ 𝛿(𝐺 𝐼 (𝑁)).

Where 𝛿(𝐺 𝐼 (𝑁)) = minimum degree of the vertex in 𝐺 𝐼 (𝑁).

Prime ideal and Ideal symmetry

Ideal symmetric graph: The graph (𝐺 𝐼 (𝑁))is said to be ideal symmetric if for every pair of vertices 𝑥, 𝑦 in (𝐺 𝐼 (𝑁)) with an edge between them, either 𝑑𝑒𝑔(𝑥) = 𝑑𝑒𝑔(0) or, 𝑑𝑒𝑔(𝑦) = 𝑑𝑒𝑔(0).

Example: (𝐺 𝐼 (ℤ 4 )) is the ideal symmetry graph, where 𝐼 = {0, 2}𝑚𝑜𝑑(4) and ℤ 4 = {0,1,2,3}𝑚𝑜𝑑(4).

Proposition 2.1:

The ideal symmetry of 𝐺 𝐼 (𝑁) implies the symmetry determine by the automorphism group of 𝐺 𝐼 (𝑁).

Proposition 2.2:

Let 𝑁 be zero symmetric near-ring and 𝐼 be a 3semi-prime ideal of 𝑁 and If 𝐼 is a strong vertex cut 𝐺 𝐼 (𝑁) then, 𝐺 𝐼 (𝑁)is an ideal symmetric graph.

Proof: Suppose 𝑁 be zero-symmetric near-ring and 𝐼 be a 3semi prime ideal of 𝑁. Also if 𝐼 be a strong vertex cut of 𝐺 𝐼 (𝑁). So, 𝐼 is a 3prime ideal of 𝑁. Hence 𝐺 𝐼 (𝑁) is an ideal symmetric graph.

Proposition 2.3:

Let 𝑁 be a commutative ring with unit element 1. Let 𝐼 be completely semi-prime ideal of 𝑁. If 𝐺 𝐼 (𝑁)is ideal symmetric, then 𝐼 is completely -prime and 𝐼 is a strong vertex cut of 𝐺 𝐼 (𝑁).

1. Anderson FW, Fuller KR. Rings and Categories of Modules, 2 nd Edition, Springer-Verlag, New York, 1992.
2. Bhavanari S, Vijyakumari. Graph a near-ring with respect to an ideal, Comm. Algebra, 2010; 38:1957-1967.
3. Bondy JA, Murty USR. Graph theory with applications, 5 th Edition, The Macmillan Press Ltd, London, 1982.
4. Booth GL, Groenewald NJ, Veldsman S. A Kurosh-Amitsur prime radicals for near rings, Comm. Algebra, 1990; 18(9):3111-3122.
5. Ferrero CC, Ferrero G. Near-rings: Some developments linked to Semi-groups and groups, Kluwer Academic Publishers, Amsterdam, Netherland, 2002.
6. Groenewald NJ. The completely prime radical in near-rings, Acta Math. Hungar, 1988; 51:301-305.
7. Groenewald NJ. Different prime ideals in near-rings, Comm. Algebra, 1991; 19:2667-2675.
8. Veldsman S. On equi-prime near-rings, Comm. Algebra, 1992; 20(9):2569-2587.