Article Search
eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2016; 56(4): 1103-1113

Published online December 23, 2016

### On the Diameter, Girth and Coloring of the Strong Zero-Divisor Graph of Near-rings

Prohelika Das

Department of Mathematics, Cotton College State University, Guwahati 781001, Assam, India

Received: May 8, 2014; Accepted: January 13, 2015

### Abstract

In this paper, we study a directed simple graph Γs(N) for a near-ring 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, JV*(N), I is adjacent to J if and only if IJ = 0. Here, we deal with the diameter, girth and coloring of the graph Γs(N). Moreover, we prove a sufficient condition for occurrence of a regular element of the near-ring N in the left annihilator of some vertex in the strong zero-divisor graph Γs(N).

Keywords: Near-ring, N-subsets, diameter, girth, essential ideal, chromatic number, left annihilator.

### 1. Introduction

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 NII(INI) 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) = {xN | xI = 0} is the left annihilator of I. The set Zl = {nN | for some xN{0}, nx = 0} [12] is the set of left zero-divisors of N. We consider the strong zero-divisor graph Γs(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, JV*(N), I is adjacent to J if and only if IJ = 0. If I and J are singleton sets, then the strong graph Γs(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 ΓI (R) with vertices {xRI | xyI for some yRI} where distinct vertices x and y are adjacent if and only if xyI. 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 dN is distributive if d(n1 + n2) = dn1 + dn2 for any n1, n2N and Nd denotes the set of all distributive elements of N. If N = Nd and (N, +) =< Nd >, 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 Γs(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 Γs(N) has a cycle with an invariant vertex, then grs(N)) ≤ 4.

Moreover, in this paper, we deal with coloring of Γs(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 Γs(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, IA ≠ 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 Γs(N) having bipartite character, i.e., the set of vertices of Γs(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 Γs(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 Γs(N) is bipartite with nonzero nilpotent invariant subsets in N, then Γs(N) is a star graph.

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

### Example 1.1

• Γs(Z4) ≅ Γ(Z4)

• Γs(Z2× Z2) ≅ Γ(Z2× Z2) ≅ Γs(Z6) (Z2× Z2Z6)

• $Γs(Z3[x]〈x2〉)≅Γs(Z2[x,y]〈x2,y2,xy〉)$ 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 Γs(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 Γs(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) = {xN | 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 Jl(I) is a nilpotent left N-subset of N, then l(I + J) = 0.

Proof

Since l(I) ≠ 0, there exists an x(≠ 0) ∈ N such that xI = 0. Now J is nilpotent gives a positive integer m such that xJm = 0 and xJm−1 ≠ 0. Again xJm−1J = xJm = 0 and xJm−1I = xJm−2JI = 0. Thus xJm−1(I + J) = 0 giving thereby xJm−1l(I + J). Thus l(I + J) ≠ 0.

Thus in this lemma, we see that the nilpotency of Jl(I) leads us to l(I+J) ≠ 0. In the next, we present diameter of the strong zero-divisor graph Γs(N), where N is a strongly semi-prime near-ring.

### 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 diams(N)) = 3.

Proof

We give the proof in two steps such as

• (i) Step1:

Suppose d(I, J) = 2. Let MV*(N) be such that, IMJ is a directed path. Then IM = 0 and MJ = 0 which gives that Mr(I) = l(I). Now, M(I + J) = 0 gives that M(≠ 0) ⊆ l(I + J). Thus l(I + J) ≠ 0, a contradiction.

• (i) Step2:

CaseI: If IJ ≠ 0, consider M = l(I), N = l(J). Claim: IM = l(I) → N = l(J) → J is a directed path. It is enough to show that l(I)l(J) = 0. Suppose there exists an xl(I), yl(J) such that xy ≠ 0. Now xl(I) = r(I) gives Ixy = 0. Thus xyr(I) = l(I) gives xyI = 0. Again yl(J) gives xyJ = 0. Now we get xy(I + J) = 0 which gives (0 ≠)xyl(I + J), a contradiction.

CaseII: If IJ = 0, then (I+J)2I2+J2. And l(I+J)2 = 0, as xl(I+J)2 gives x(I+J) ⊆ l(I+J) giving thereby xl(I+J) = 0. Since J is nilpotent, qJ is also so where ql(I) with qJ2 ≠ 0. Now qJl(I) gives l(I +qJ) ≠ 0 [Lemma 2.2]. Again I +qJ = J, otherwise IJ implies l(I +J) = l(J) ≠ 0, a contradiction. Hence I + qJ,J are distinct and I + J = I + qJ + J which gives l(I + qJ + J) = 0 and (I + qJ)J ≠ 0. Hence d(I + qJ, J) = 3[caseI].

### 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 Jl(I1) with l(I1) ∩ l(J) = 0.

Proof

Let yl(I1 + J), y = ∑±di where l(I1 + J) =< S >, diS, a set of distributive elements of N. Now di(I1 + J) = 0 gives (diI1 + diJ)i1 = 0 for each i1I1. Thus diI1i1 = 0 as Jl(I1) giving thereby dil(I1) = l(I1i1), since l(I1) is maximal. Now we get diJ = 0 which gives dil(I1)∩l(J) for each i. Thus y = 0 which gives l(I1 +J) = 0 giving thereby l(I +J) = 0. Hence d(I, J) ≠ 2. [Theorem 2.3(i)]

### Theorem 2.5

Let P1 = l(I1) and P2 = l(I2) be two prime ideals of N such that P1P2 = 0, where I1and I2are invariant subsets of N. Then I1I2 = (0) = I2I1.

Proof

For I1I2 ≠ 0, we get I1l(I2) = P2 and I2r(I1) = l(I1) = P1. Now P1I1P2 gives P1P2 as I1P2 = l(I2) giving thereby P1P2 = P1 ≠ 0, a contradiction. Similarly, I2I1 = 0

### Definition 2.6

Invariant associated of a near-ring N denoted by IAss(N) is the collection of l(Ii)’s, where each l(Ii) is a prime ideal with invariant N-subset Ii such that l(Ii) ∩ l(Ij) = 0 for ij.

### Corollary 2.7

If in a strongly semi-prime near-ring N, |(IAssN)| ≥ 3, then grs(N)) = 3.

Proof

Let IAss(N) = {P1, P2, P3}, then P1 = l(I1), P2 = l(I2) and P3 = l(I3) for some invariant subsets I1, I2 and I3 respectively. Then I1I2 = 0, I2I3 = 0 and I3I1 = 0 [theorem 2.5]. Hence I1I2I3I1 is a cycle of length 3. Thus grs(N)) = 3.

### Theorem 2.8

If |IAssN| ≥ 5, then Γs(N) is not a planner graph.

Proof

Let IAssN = {P1, P2, P3, P4, P5} where Pi = l(Ii)(say), 1 ≤ i ≤ 5. Here IiIj = 0 for ij [theorem 2.5]. Thus the graph Γs(N) contains Kuratowski’s first graph. Hence Γs(N) is not planner.

Next we determine the girth of the graph Γs(N) of a stongly semi-prime near-ring N if it has a cycle with at least one invariant vertex.

### Theorem 2.9

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

Proof

Assume n = grs(N)) is 5, 6 or 7. Let I1I2I3..... → InI1...(i) be a cycle with minimal length n. Let Ii be an invariant vertex. Now consider the subgraph $Γs/(N)$ of Γs(N) spanned by the vertices I1, I2, ...., IiIi+2. If IiIi+2Ik for any k, 1 ≤ kn, then Ii−1IiIi+1IiIi+2Ii−1, (i ≥ 2) is a cycle of length 4. Let IiIi+2 = Ik for some k. Now we show the following.

• (i) IiIi+2Ii+1. If IiIi+2 = Ii+1, then (IiIi+2)Ii−1 = Ii+1Ii−1. NowIi+1Ii−1 = (IiIi+2)Ii−1IiIi−1 = 0, which gives Ii+1Ii−1 = 0. Thus Ii−1IiIi+1Ii−1 is a cyclic, a contradiction to (i).

• (ii) IiIi+2Ii−1. For otherwise, Ii+1(IiIi+2) = Ii+1Ii−1 which gives Ii+1Ii−1 = (Ii+1Ii)Ii+2 = 0. Thus Ii−1IiIi+1Ii−1 is a cycle, a contradiction to (i).

• IiIi+2Ii+3. If IiIi+2 = Ii+3, then we get Ii+3Ii+1 = (IiIi+2)Ii+1IiIi−1 = 0, gives the cycle Ii+1Ii+2Ii+3Ii+1, a contradiction.

Now IiIi+2 is adjacent to three distinct vertices Ii−1, Ii+1 and Ii+3. Thus there exists an extra edge in $Γs/(N)$ which is not in the original cycle. Hence there must exist a smaller cycle $Γs/(N)$, a contradiction.

Now we present coloring of the strong zero-divisor graph Γs(N) of N.

### Theorem 2.10

Let N be a strongly semi-prime near-ring. If N has k number of maximal ideals of the form l(Ii) where Ii’s are invariant subsets such that l(Ii) ∩ l(Ij) = 0 for ij, 1 ≤ i, jk, then χs(N)) ≤ k + 1.

Proof

First we give k distinct colors to Ii’s and an extra color to 0. Here IiIj = 0 for ij [Theorem 2.6]. Now we color the invariant vertices. If I(≠ 0) be an arbitrary invariant vertex, we give to I the color which is given to $Inth$ vertex, where n is the minimal {i|l(I) ⊈ l(Ii)}. Let I and J be two invariant vertices such that same color of Ik is given to them. Then l(I) ⊈ l(Ik) and l(J) ⊈ l(Ik). If IJ = 0, then Il(J) ⊈ l(Ik) and Jr(I) = l(I) ⊈ l(Ik) which leads to IJl(Ik), a contradiction. Next we show that these k + 1 colors are enough to color the whole graph. Let I(≠ 0) be a left N-subset of N and IV*(N). Consider Il(J)(≠ 0) with some JV*(N). If Il(J) = 0 for any JV*(N), then Il(In) = 0 for all n, 1 ≤ nk. Thus l(In) ⊆ l(I) gives l(In) = l(I) for all n, a contradiction. Now we give the color to I which is given to the invariant vertex Il(J). Here IIl(J) ≠ 0, for otherwise Il(Il(J)) = r(Il(J)) which gives Il(J)I = 0, giving thereby (Il(J))2 = 0, a contradiction. Suppose I and I/ has the color of Ik(say). Then we get some J, J/V*(N) such that Il(J) and I/l(J/) are given the color of Ik. Nowl(Il(J)) ⊈ l(Ik) and l(I/l(J/)) ⊈ l(Ik). If (Il(J))(I/l(J/)) = 0, then Il(J) ⊆ l(I/l(J/)) ⊈ l(Ik) and I/l(J/) ⊆ r(Il(J)) = l(Il(J)) ⊈ l(Ik) which implies that (Il(J))(I/l(J/)) ⊈ l(Ik), a contradiction. Now we show that I and I/ are not adjacent. If II/ = 0, then II/l(J/) = 0 gives (I/l(J/))I = 0. Thus (I/l(J/))(Il(J)) = 0 gives (Il(J))2 = (I/l(J/))2 = 0, a contradiction.

### 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 χs(Z6)) is 2 + 1 = 3, i.e., χs(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 Γs(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.

Proof
• (i) ⇒ (ii) Suppose J is an invariant N-subset of N such that J2 = 0. Let A be a nonzero ideal of N. If AJ = 0 then A = Al(J) ≠ 0. If AJ ≠ 0, then AJ(≠ 0) ⊆ Al(J). Thus in either cases l(J) is essential. Hence J = 0.

• (ii) ⇒ (i) Let I be a left N-subset such that l(I) is essential. Let J = l(I)∩IN. Now J2l(I)IN = 0. Thus J = 0, i.e., l(I)∩IN = 0 which gives IN = 0 as l(I) is essential. Hence I = 0.

### 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 xN 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.

Proof

Let I1 < I2 < I3 < ......be an ascending chain for left N-subsets, where each Ii’s are essential in N. Suppose Ii < Ii+1. Now Iil(Ii) < Ii+1l(Ii). Here Iil(Ii) ≠ 0 and Ii+1l(Ii) = 0. Also Iil(Ii) ≠ Ii+1l(Ii) for otherwise (Iil(Ii))2 = (Ii+1l(Ii))(Iil(Ii)) ⊆ l(Ii)Ii = 0, a contradiction. Now consider an element xnInl(In−1) such that xnIn−1l(In−1). Here for i = j (suppose i > j), xixj ∈ (Iil(Ii−1))(Ijl(Ij−1)) ⊆ l(Ii−1)Ij = 0. Thus we get an infinite clique in N, a contradiction.

### Theorem 2.17

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

Proof

Let I1 < I2 < I3.... be an ascending chain of invariant subsets with essential left annihilators. Suppose IiIi+1. Let xi+1(≠ 0) ∈ Ii+1Ii. Now consider Ji+1 = l(Ii+1) ∩ 〈xi+1〉 ≠ 0, where 〈xi+1〉 is the ideal generated by xi+1. Here JiJj = 0 for i < j, a contradiction.

### Theorem 2.18

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

Proof

We give n distinct colors to l(Ii)’s. Here IiIi+1 = 0 since for otherwise (l(Ii)∩IiIi+1) ≠ 0. Now(l(Ii)∩IiIi+1)2l(Ii)IiIi+1 = 0 which gives l(Ii)∩IiIi+1 = 0, a contradiction. Now let I be an arbitrary vertex.

• (i) CaseI: If IiI for some i, then give the color of Ik to I if k is the max{i|IiI}. Here I and Ik are not adjacent since for otherwise Il(Ik) together with IkI gives that (Ik)2 = 0, a contradiction.

• (ii) CaseII: If IiI for any i, then there exists an xIi such that xI. Now consider the ideal generated by x denoted < x > which is clearly non zero. Thus l(Ii)∩ < x >≠ 0. But (l(Ii)∩ < x >)2l(Ii)Ii = 0, a contradiction.

Suppose two distinct vertices I and J are given the same color of Ik(say). Here IJ = 0 for otherwise Il(J) which leads IkIl(J). Thus we get $Ik2=0$ as IkJ, a contradiction.

Now we mention the following notes:

• (i) Note 1: In a near-ring N, χs(N)) = 2 if and only if for any two nonzero I, JV*(N), IJ ≠ 0 whenever I ≠ 0, J ≠ 0. For, suppose there exists I ≠ 0 and J ≠ 0 such that IJ = 0. Then {0, I, J} is a clique. Thus clique(Γs(n)) > χs(N)), a contradiction.

• (ii) Note 2: In a strongly semi-prime near-ring without unity, every essential ideal of the form l(Ii) with invariant Ii is maximal. For suppose l(Ii) is not maximal, there exists a proper ideal K of N such that l(Ii) ⊂ KN. Now consider the ideal J generated by Iix(≠ 0) for some x(≠ 0) ∈ K. Here l(Ii) ∩ J = 0 but (l(Ii) ∩ J)2 = 0, a contradiction.

### 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, bZ(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 Ii’s are essential Zp is a reduced ring without unity. But here l(Ik−1) = 0 for all k which are not essential. Here IiIj ≠ 0 for any i, j and χs(Zp)) = 2.

### Example 2.20

Consider the set $M(N)=(Z2N0Z2)$ which is the set of elements of the form { $(0n00),(0n01),(1n00),(1n01)$}, where nN. Here M(N) is a near-ring with respect to ordinary addition and multiplication(χ̄n = xn, χ̄Z2) with unity which is not strongly semi-prime as $(0n00)·(0n00)=(0000)$. If N is not finite, then M(N) has infinite invariant sets Ii(i = 1, 2, 3, ...) such that $l(Ii)={(0n00),(0000)∣n∈N}$

is essential and χs(M(N))) = ∞.

### Theorem 2.21

Let Γs(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 Γs(N) is a star graph with more than one vertices.

Proof
• (i) Let I1, I2 and I3 be three distinct invariant N-subsets of N such that l(Ii)’s are essential. Now J1 = l(I1)∩I2 ≠ 0, J2 = l(I3)∩I2 ≠ 0 and J3 = l(I2)∩I3 ≠ 0. Here J3J1 for otherwise (l(I2) ∩ I3)2 = (l(I2) ∩ I3)(l(I1) ∩ I2) = 0, a contradiction. Thus (l(I2) ∩ I3)(l(I1) ∩ I2) = 0. Similarly (l(I1) ∩ I2)(l(I3) ∩ I1) = 0 and (l(I3) ∩ I1)(l(I2) ∩ I3) = 0. Thus J1J2J3J1 is a cycle, a contradiction.

• (ii) Suppose N is not strongly semi-prime and let I ≠ 0 be an invariant N subset such that I2 = 0. Assume that IV1. We show that V1 = {I}. Here either l(I) is not essential or I is minimal. Suppose l(I) is essential and there exists an I1(≠ 0) such that I1I. Now l(I) ∩ I1 ≠ 0 and (l(I) ∩ I1)I = 0 gives l(I)∩I1V2 and II1 = 0 gives I1V2. But (l(I)∩I1)I = 0, a contradiction. Now we consider the following cases.

• (a) CaseI: If l(I) is essential. Suppose there exists a PV1 {I}. If IP = 0, then PV2, a contradiction. Hence IP ≠ 0. Since Γs(N) is connected, there exists a KV2 such that PK = 0. Now l(I)∩IP ≠ 0 and I(l(I) ∩ IP) = 0 gives that l(I) ∩ IPV2. But (l(I) ∩ IP)K = 0, a contradiction.

• (b) CaseII: If l(I) is not essential. Now suppose I is minimal. Then IP = P which gives that (IP)I = I2 = 0. Thus IPV2. But (IP)K = 0, a contradiction. If I is not minimal, then IPI which gives (IPI)K = (IP)K = 0. Thus IPIV2, a contradiction to (IPI)K = 0.

### 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 Γs(N) such that every other vertex is adjacent to I. Then l(I) contains a left non-zero divisor.

Proof

Here l(I) is essential. Consider I1(≠ 0) ⊆ l(I) such that I1 is non nilpotent and l(I1) is as large as possible. If l(I1) = 0, we stop. If not, there exists a left N-subset X(≠ 0) such that XI1 = 0. But Xl(I) ≠ 0 as l(I) is essential. Consider a1(≠ 0) ∈ Xl(I) such that l(Na1) is as large as possible. Now Na1Xl(I). If l(Na1) = 0, we stop. Suppose l(Na1) ≠ 0. Now Na1I1l(I) and Na1 + I1l(I). If l(Na1 + I) = 0, then we stop. If not, then l(Na1 + I1) ∩ l(I) ≠ 0. Again l(Na1 + I1) = l(Na1) ∩ l(I1)[theorem 2.4] gives l(Na1 + I1) ∩ l(I) = l(Na1) ∩ l(I1) ∩ l(I) ≠ 0. Now we consider a2(≠ 0) ∈ l(Na1) ∩ l(I1) ∩ l(I) with a2 non nilpotent and l(Na2) is as large as possible. If l(Na2 + Na1 + I1) = 0, we stop. If not, proceeding in the same way, we get l(I1) ⊇ l(I1) ∩ l(Na1) ⊇ l(I1) ∩ l(Na1) ∩ l(Na2) ⊇ ....... which is stationary. Hence we get a positive integer t such that l(I1) ∩ l(Na1) ∩ .... ∩ l(Nat) = l(I1) ∩ l(Na1) ∩ .... ∩ l(Nat+1). Now l(I1)+l(Na1)+....+l(Nat) = l(I1)+l(Na1)+....+l(Nat+1)=l(I1+Na1+....+Nat)∩ l(Nat+1) ⊆ l(Nat+1). Now Nat+1l(I1+Na1+....+Nat) gives Nat+1l(Nat+1) which gives (Nat+1)2 = 0, a contradiction. Thus l(I1 +Na1 +....+Nat) ∩ l(I) = 0 giving thereby l(I1 + Na1 + .... + Nat) = 0 as l(I) is essential.

### Acknowledgement

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

### References

1. Akbari, S, and Mohammadian, A (2004). On the zero-divisor graph of a commutative ring. J Algebra. 274, 847-855.
2. Anderson, DD, and Naser, M (1993). Beck’s Coloring of a commutative ring. J Algebra. 159, 500-541.
3. Anderson, DD, and Livingston, PS (1999). The zero-divisor graph of a commutative ring. J Algebra. 217, 434-447.
4. Anderson, DD, and Mulay, SB (2007). On the diameter and girth of a zero-divisor graph. J Pure Appl Algebra. 210, 543-550.
5. Beck, I (1988). Coloring of commutative rings. J Algebra. 116, 208-226.
6. Behboodhi, M, and Rakeei, Z (2011). The annihilating ideal graph of commutative ring. J Algebra Appl. 10, 727-739.
7. Chowdhury, KC, and Saikia, H (1997). On near-ring with ACC on annihilators. Mathematica Pannonica. 8, 177-185.
8. Diestel, R (1997). Graph Theory. New York: Springer-Verlag
9. Pilz, G (1977). Near-rings: North Holland Publishing Company
10. Redmond, SP (2003). An ideal-based zero-divisor graph of a commutative ring. Comm Algebra. 31, 4425-4443.
11. Tamuli, BK, and Chowdhury, KC (1988). Goldie Near-rings. Bull Cal Math Soc. 80, 261-269.
12. Wendt, G (2009). On Zero-divisors in Near-Rings. International Journal of Algebra. 3, 21-32.