In this paper by a near-ring N, we mean a zero symmetric (right) near-ring not necessarily containing 1. A subset I of N is left(right)N-subset of N if NI ⊆ I(IN ⊆ I) and I is invariant if it is both left as well as right N-subset of N. If I is a left N-subset of N, then the ideal l(I) = {x ∈ N | xI = 0} is the left annihilator of I. The set Z_l = {n ∈ N | for some x ∈ N{0}, nx = 0} [12] is the set of left zero-divisors of N. We consider the strong zero-divisor graph Γ_s(N), where the set V^*(N) of vertices is the set of all left N-subsets of N with nonzero left annihilators and for any two distinct vertices I, J ∈ V^*(N), I is adjacent to J if and only if IJ = 0. If I and J are singleton sets, then the strong graph Γ_s(N) reduced to the graph Γ(N) of N where x(≠ 0) ∈ N is adjacent to y(≠ 0) ∈ N if and only if xy = 0.

The concept of zero-divisor graph of a commutative ring was first introduced by Beck in [5]. Beck [5] was mainly interested in the coloring of the ring. This notion was redefined in [3] and they proved that such a graph is always connected and its diameter is always less than or equal to 3. Anderson and Mulay in [4] studied diameter and girth of zero-divisor graph of a commutative ring. The notion of zero-divisor graph was extended to a non-commutative ring [1] and various properties of diameter and girth were established. In [10], Redmond has generalised the notion of zero-divisor graph. For an ideal I of a commutative ring R, Redmond [10] defined an undirected graph Γ_I (R) with vertices {x ∈ RI | xy ∈ I for some y ∈ RI} where distinct vertices x and y are adjacent if and only if xy ∈ I. Behboodhi [6] studied annihilator ideal graph dealing with the annihilators of ideals of a commutative ring.

In this paper, we study the graph theoretic aspect of a near-ring N which is a less symmetric algebraic structure with + and ., where both operations are noncommutative. An element d ∈ N is distributive if d(n₁ + n₂) = dn₁ + dn₂ for any n₁, n₂ ∈ N and N_d denotes the set of all distributive elements of N. If N = N_d and (N, +) =< N_d >, then N is distributive and distributively generated, respectively. For a distributive near-ring N with 1, the graph Γ(N) is the zero-divisor graph of a non-commutative ring N.

For basic definitions and results related to near-ring, we would like to mention Pilz [9].

Recall that a graph G is connected if there is a path between any two distinct vertices and is complete if every two vertices are adjacent. The distance between two distinct vertices x and y of G is the length of the shortest path from x to y and is denoted by d(x, y). If no such path exists, then d(x, y) = ∞. The diameter of the graph G is the sup{d(x, y)|x and y are distinct vertices of G} and is denoted by diamG. The girth of G is the length of distance of the shortest cycle in G, denoted by gr(G). If no such cycle, then gr(G) = ∞.

A left N-subset I of N is nilpotent if there exists a positive integer n such that Iⁿ = 0 and Iⁿ⁻¹ ≠ 0. The near-ring N is strongly semi-prime if it has no nonzero nilpotent invariant subsets. The notion of simple graph excludes the loops which is compatible to the strongly semi-prime character of the near-ring. The graph that we dealt here is a connected one and has diameter 3 or less, the proof of which follows in alike way to that of the theorem 2.3 [3]. It is due to the proposition 1.3.2 [8], if a graph G has a cycle, then the gr(G) is less than 2diamG + 1. In this paper, we study diameter and girth of the strong zero-divisor graphs of near-rings. Anderson [3] has conjectured that if a zero-divisor graph had a cycle, then its girth was 3 or 4. Haevey Mudd and Jamson gave an elegant proof to the conjecture of Anderson[3]. We establish a sufficient condition for diameter 3 for the graph Γ_s(N) of the near-ring N. Existence of a cycle in the strong zero-divisor graph deserves exclusive interest. We prove that in a strongly semi-prime near-ring, if Γ_s(N) has a cycle with an invariant vertex, then gr(Γ_s(N)) ≤ 4.

Moreover, in this paper, we deal with coloring of Γ_s(N). The minimal numbers of colors so that no two adjacent elements of the graph G have same color is the chromatic number of G and is denoted by χ(G). A clique of G is the maximal connected subgraph of it. The number of vertices in the largest clique in the graph G is the cliqueG. Beck, [5] conjectured that χ(Γ(R)) = clique(Γ(R)). But D.D.Anderson and M.Nasser [2] gave the counter example such as R = Z₄[[x, y, z]/(x² − 2, y² − 2, z², 2x, 2y, 2z, xy, xz, yz − 2)] for which χ(Γ(R)) = 5 and clique(Γ(R)) = 4. Beck [5]has proved characterisation of rings with finite chromatic number and showed that such rings have the ascending chain condition(acc) on annihilators. We here deal with the strong zero-divisor graph Γ_s(N) having finite chromatic number. A left N-subset(ideal) I of N is essential in N if for any non zero left N-subset(ideal) A of N, I ∩ A ≠ 0. We prove that chromatic number of such a graph showing alike relation with the numbers of maximal annihilator ideals as well as with that of essential annihilator ideals of the near-ring. Also we deal with the strong zero-divisor graph Γ_s(N) having bipartite character, i.e., the set of vertices of Γ_s(N) can be decomposed into two disjoint parts such that every edge joints a vertex of one part to that of the other part. We establish that if Γ_s(N) is bipartite where N is strongly semi-prime without unity, then N has exactly two invariant subsets I₁ and I₂ (say) provided l(I₁) and l(I₂) are essential. In addition to it we show that if Γ_s(N) is bipartite with nonzero nilpotent invariant subsets in N, then Γ_s(N) is a star graph.

The following are some examples of strong zero-divisor graphs.

Example 1.1

• Γ_s(Z₄) ≅ Γ(Z₄)

• Γ_s(Z₂× Z₂) ≅ Γ(Z₂× Z₂) ≅ Γ_s(Z₆) (Z₂× Z₂ ≇ Z₆)

• Γs(Z3[x]〈x2〉)≅Γs(Z2[x,y]〈x2,y2,xy〉) but Z3[x]x2≇Z2[x,y]〈x2,y2,xy〉

• Γs(Z4[x]〈x2〉)≇Γs(Z2[x,y]〈x2,xy,y2〉) (Z4[x]〈x2〉≅Z2[x,y]〈x2,xy,y2〉)

2. Main Results

In this section, we present results regarding diameter and girths of Γ_s(N) in contrast to Γ(N) in some cases. We note that the vertex 0 is adjacent to every other vertices which we exclude here for obvious reason.

A vertex I of Γ_s(N) is an invariant vertex if it is an invariant N subset of the near-ring N. The right annihilator of a left N-subset I of N is r(I) = {x ∈ N | Ix = 0} which is a right N-subset of N, need not coincide to l(I) in general. However in a strongly semi-prime near-ring N for an invariant subset I, Il(I) = 0 as (Il(I))² = I(l(I)I)l(I) = 0 giving thereby l(I) ⊆ r(I). Similarly r(I) ⊆ l(I). Thus we state the following lemma.

Lemma 2.1

Let N be a strongly semi-prime near-ring. Then for an invariant subset I of N, l(I) = r(I).

For a subset I of N, l(I) ≠ 0 may not imply l(I + J) ≠ 0 for any subset J of N. Below we present when it occurs.

Lemma 2.2

Let N be a near-ring such that the left annihilators are distributively generated. If I be a left N-subset with l(I) ≠ 0 and J ⊆ l(I) is a nilpotent left N-subset of N, then l(I + J) = 0.

Theorem 2.3

Let N be a strongly semi-prime near-ring such that the left annihilators are distributively generated. If there exists a nilpotent vertex J and an invariant vertex I such that l(I + J) = 0, then diam(Γ_s(N)) = 3.

Theorem 2.4

Let N be a strongly semi-prime near-ring such that the left annihilators are distributively generated. If I is an invariant N-subset of N containing a non nilpotent subset I₁with maximal left annihilators, then d(I, J) ≠ 2 for any J ⊆ l(I₁) with l(I₁) ∩ l(J) = 0.

Theorem 2.5

Let P₁ = l(I₁) and P₂ = l(I₂) be two prime ideals of N such that P₁ ∩ P₂ = 0, where I₁and I₂are invariant subsets of N. Then I₁I₂ = (0) = I₂I₁.

Definition 2.6

Invariant associated of a near-ring N denoted by I − Ass(N) is the collection of l(I_i)’s, where each l(I_i) is a prime ideal with invariant N-subset I_i such that l(I_i) ∩ l(I_j) = 0 for i ≠ j.

Corollary 2.7

If in a strongly semi-prime near-ring N, |(I − AssN)| ≥ 3, then gr(Γ_s(N)) = 3.

Theorem 2.8

If |I − AssN| ≥ 5, then Γ_s(N) is not a planner graph.

Theorem 2.9

Let N be a strongly semi-prime near-ring. If Γ_s(N) contains a cycle with an invariant vertex in it, then gr(Γ_s(N)) ≤ 4.

Theorem 2.10

Let N be a strongly semi-prime near-ring. If N has k number of maximal ideals of the form l(I_i) where I_i’s are invariant subsets such that l(I_i) ∩ l(I_j) = 0 for i ≠ j, 1 ≤ i, j ≤ k, then χ(Γ_s(N)) ≤ k + 1.

Example 2.11

Consider Z₆ = {0, 1, 2, 3, 4, 5} which is a near-ring with respect to the tables given below. The only left N subsets are I₁ = {0, 3}, I₂ = {0, 2, 4} and I₃ = {0, 2, 3.4} which are invariant also and l(I₁) = I₂ and l(I₂) = I₁ are two maximal ideals of the annihilator ideal form. Here the chromatic number χ(Γ_s(Z₆)) is 2 + 1 = 3, i.e., χ(Γ_s(Z₆)) is equal to p + 1, where p is the number of maximal ideals of the form of left annihilator.

In the results below, we deal with the essentiality of annihilator ideals in a near-ring N to determine the chromatic number of Γ_s(N).

Theorem 2.12

Let N be a near-ring with unity, then the following two are equivalent.

• (i) If for a left N-subset I of N, l(I) is essential, then I = 0.

• (ii) N is strongly semi-prime.

Example 2.13

Consider the ring Z₆ = {0, 1, 2, 3, 4, 5} which is strongly semi-prime with unity. Here I₁ = l(I₂) = {0, 3} and I₂ = l(I₁) = {0, 2, 4} are the only nonzero ideals and Z₆ = Ann(0) is the only essential ideal.

Example 2.14

Z₄ = {0, 1, 2, 3} is a ring with unity. Here Z₄ is not strongly semi-prime as for I = {0, 2}, I₂ = 0 and l(I) is an essential ideal of Z₄

Theorem 2.15

Let N be a near-ring and x ∈ N be such that every vertex v ∈ Γ(N) is adjacent to x. Then l(x) is an essential ideal of N.

Theorem 2.16

Let N be a strongly semi-prime near-ring. If Γ(N) has no infinite clique, then the near-ring N satisfies the acc on essential left N-subsets.

Theorem 2.17

Let N be a strongly semi-prime near-ring without unity. If Γ_s(N) has no infinite clique, then N satisfies the acc on invariant subsets having essential left annihilators.

Theorem 2.18

Let N be a strongly semi-prime near-ring without unity and l(I₁), l(I₂), ...., l(I_n) are the only essential N-subsets of N with each I_iis an ideal. Then χ(Γ_s(N)) ≤ n + 1.

Example 2.19

Consider the set Z_(p^∞) of all rational numbers of the form mpk such that 0≤mpk<1 , where p is a fixed prime number, n runs through all nonnegative integers. Then Z(p^∞) is a ring with respect to addition modulo 1 and multiplication defined as ab = 0 for all a, b ∈ Z(p^∞). It is to be noted that each subgroup of Z(p^∞) is an ideal of it and the only proper ideals of Z(p^∞) are of the form Ik-1={0,1pk-1,1pk-2,....,pk-1-1pk-1} for each positive integer k. Thus the ideals are in a chain 0 < I₁ < I₂ < ..... and each I_i’s are essential Z_p^∞ is a reduced ring without unity. But here l(I_k−1) = 0 for all k which are not essential. Here I_iI_j ≠ 0 for any i, j and χ(Γ_s(Z_p^∞)) = 2.

Example 2.20

Consider the set M(N)=(Z2N0Z2) which is the set of elements of the form { (0n00),(0n01),(1n00),(1n01)}, where n ∈ N. Here M(N) is a near-ring with respect to ordinary addition and multiplication(χ̄n = xn, χ̄ ∈ Z₂) with unity which is not strongly semi-prime as (0n00)·(0n00)=(0000). If N is not finite, then M(N) has infinite invariant sets I_i(i = 1, 2, 3, ...) such that l(Ii)={(0n00),(0000)∣n∈N}

is essential and χ(Γ_s(M(N))) = ∞.

Theorem 2.21

Let Γ_s(N) be a bipartite graph with two non-empty parts V₁and V₂. Then

• (i) If N is strongly semi-prime without unity, then N has exactly two invariant N subsets I₁and I₂, where l(I₁) and l(I₂) are essential.

• (ii) If N is not strongly semi-prime, then Γ_s(N) is a star graph with more than one vertices.

Theorem 2.22

([7]) A strongly semi-prime near-ring N satisfying the acc on left annihilators has no nonzero nil left N-subsets in it.

In the example 2.11, we see that every essential left ideal is essential as left N-subgroup also. We call such a near-ring a near-ring with total essential character. Moreover for near-ring with the a.c.c on annihilators, we would like to refer [11].

Theorem 2.23

[7]) If a strongly semi-prime near-ring N is with total essential character, then N satisfies the dcc (descending chain condition) on left annihilators.

Theorem 2.24

Let N be a strongly semi-prime near-ring with the acc on left annihilators satisfying total essential character and the left annihilators are distributively generated. Let I be a vertex of Γ_s(N) such that every other vertex is adjacent to I. Then l(I) contains a left non-zero divisor.

I would like to acknowledge the referees for their valuable suggestions.