Essential Ideal of a Matrix Nearring and Ideal Related Properties of Graphs

: In this paper, we consider matrix maps over a zero-symmetric right nearring N with 1. We deﬁne the notions of f -essential ideal, f -superﬂuous ideal, generalized f -essential ideal of a matrix nearring and prove results which exhibit the interplay between these ideals and the corresponding ideals of the base nearring N . We discuss the combinatorial properties such as connectivity, diameter, completeness of a graph (denoted by L g ( H )) deﬁned on generalized essential ideals of a ﬁnitely generated module H over N . We prove a characterization for L g ( H ) to be complete. We also prove L g ( H ) has diameter at-most 2 and obtain related properties with suitable illustrations.


Introduction
Nearring is a classical generalization of a ring.Rings can be considered as algebraic systems of linear maps on groups, while nearrings describe a general non-linear case [23].In this paper, we consider a zero-symmetric right nearring N , and matrix maps over N [19].Meldrum and Van der Walt [19] defined the notion of a matrix nearring, denoted by M n (N ), which is the subnearring of the nearring M (N n ), the set of all maps from N n to N n .Van der Walt [4] explored the relationship between primitive modules over a nearring N and those of the matrix nearring M n (N ).For recent developments in matrix nearrings, we refer to [8,9,13,14,25,27].The notion of an essential submodule of module over a ring is a discretized analogue to the notion of dense subspace in a topological space [3].The idea of the graph constructed from a ring was initiated from the concept of a zero-divisor graph (see [2]).Later, based on a ring structure, several types of graphs like annihilator essential graph (see [5]), essential graph (see [22]), total graph (see [2]), prime graph (see [11]), and in case of nearrings zero-divisor graph of a nearring [15] and graph with respect to ideal of a nearring [6,12] were studied.In commutative rings, the author (see [1]) studied the properties of an essential ideal graph and characterized rings based on the different types of graphs.They considered the set of all non-trivial ideals in a commutative ring as the vertex set and an edge is defined if the sum of two ideals is essential in the underlying ring.This concept was generalized to modules over rings by [18].In [28], small essential ideals and Morita duality of rings were discussed, and in [16], the authors characterized classes of commutative and non-commutative rings for which maximal small and minimal essential ideals coincide.In [20,21,24], the authors discussed the dual aspects like generalized supplements, superfluous ideals etc.In Section 3 of this paper, we introduce the notion of g-essential ideal in a matrix nearring.We establish a one-one correspondence between the set of all generalised essential (resp.essential) ideals of the nearring N and the set of all generalised f -essential (resp.f -essential) full ideals of M n (N ).In section 4, we define generalized essential ideal graph of a module over a nearring N (denoted as, L g (H)).We derive properties 2 R. Salvankar, K. B. Srinivas, H. Panackal and K. S. Prasad such as diameter, completeness based on a given N -group.We prove that if H is a finitely generated over a zero-symmetric nearring N , then any maximal ideal of H is a universal vertex.Furthermore, the graph is connected with the diameter less than 3.The notion of g-complement of an ideal is introduced as a generalization of a complement and show that every non-zero, non-g-essential ideal is adjacent to its g-complement.We consider a subgraph of g-essential ideal graph, induced by the set of all non-g-essential ideals of H. Finally, it is observed that there exists a path between every two superfluous ideals.

Preliminaries
Let N be a zero-symmetric right nearring and H be an N -group [23] and we write it as A ≤ e H.A uniform N -group is the one in which every non-zero ideal is essential.
then we call H is g-uniform.An N -subgroup H of N is said to be finitely generated (as an ideal) if there exists a subset S of H such that S = H, where S represents the ideal of H generated by S. We refer to [7,8,9,17,19] for the notions of essential ideals, superfluous ideals of N -groups and modules, and we refer to [23] for the notions of maximal ideal, minimal ideal and completely reducible N -group etc. Throughout, H denotes a finitely generated N -group where N is a zero-symmetric right nearring.For a zero-symmetric nearring N with 1, let N n will be the direct sum of n copies of (N, +).The elements of N n are column vectors and written as (r 1 , • • • , r n ).The symbols i i and π j respectively, denote the i th coordinate injective and j th coordinate projective maps.For an element a The nearring of n × n matrices over N , denoted by M n (N ), is defined to be the subnearring of M (N n ), generated by the set of maps {f a ij : , where f a (x) = ax, for all a, x ∈ N .If N happens to be a ring, then f a ij corresponds to the n × n-matrix with a in position (i, j) and zeros elsewhere.We refer to [9,13,19] for further definitions and notations in matrix nearrings.The graphs considered are simple graphs.We denote the vertex set as V , we use d(u, v) to represent the shortest u ∼ v path, while the eccentricity of a vertex u, is denoted as e(u) which is max{d(u, v) : v ∈ V }; radius is the minimum eccentricity, and the diameter is the maximum eccentricity.A vertex is universal if it is adjacent to every other vertex.For all other notions and definitions in graph theory, we refer to [10,9], and for nearrings, we refer to [9,23].We use ∼ to denote an edge, and ⇐⇒ for "if and only if".

Generalized essential ideals in M n (N )
In this section, we introduce the notion of generalised f -essential ideal in a matrix nearring.We establish a one-one correspondence between the set of all generalised essential (resp.essential) ideals of the nearring N and the set of all generalised f -essential (resp.f -essential) full ideals of M n (N ).Definition 3.1.[9] 1.Let K M n (N ).Then K ⋆ = {x ∈ N : x ∈ im(π j A) for some A ∈ K and j, 1 ≤ j ≤ n}.

Let I N . Then
. We use the induction on weight of A. Let w(A) = 1.
Then A = f a ij for some a ∈ N , 1 ≤ i, j ≤ n.Since N = I + J and a ∈ N , there exists x ∈ I and y ∈ J such that a = x+y.By Corollary 4.5 of [19], we get Since w(B) and w(C) is less than m, by induction hypothesis, we can write Proposition 3.7.[19] There is a bijection between the set of all ideals of N and the set of all full ideals of M n (N ) given by I → I ⋆ and S → S ⋆ such that (I ⋆ ) ⋆ = I and (S ⋆ ) ⋆ for an ideal I of N and S of M n (N ). 3.There is a one-one correspondence between the set of all essential ideals of N and the set of all essential full ideals of M n (N ). Proof.
From the Notation 3.4.6,given in [9], u = ∪A m , where m ranges from 0 to infinity, A 0 = {u} and Case 2: Let a ∈ A 0 0 .Then a = 0. Clearly Proof.To prove f u ij is an essential element in M n (N ), we need to prove Now by Lemma 3.10, we get a / ∈ u .Therefore, x ∩ u = (0).Since u ≤ e N , we get x = (0) which implies x = 0. Therefore, S ⋆ = (0).Now (S ⋆ ) ⋆ = (0) ⋆ = (0).Since S ⊆ (S ⋆ ) ⋆ , we get S = (0), which implies Definition 3.12.An ideal S of M n (N ) is said to be f -superfluous in M n (N ) if for any full ideal K of M n (N ) with S + K = M n (N ) implies K = M n (N ) and it is denoted by S ≪ f M n (N ).Lemma 3.13.

Let B be a full ideal of M
Proof.
1. Let K be a full ideal of M n (N ) such that B ⋆ + K = M n (N ).Since K is a full ideal, there exists an ideal K of N such that K ⋆ = K, which implies B ⋆ + K ⋆ = M n (N ).By Lemma 3.5, we get Since B is a full ideal, we get Definition 3.14.An ideal S of M n (N ) is called a generalized f -essential (abbr.g f -essential) ideal if for any f -superfluous full ideal T of M n (N ) S ∩ T = (0) implies T = (0).Proposition 3.15.Let S be an ideal of M n (N ) and I be an ideal of N .
3. There is a one-one correspondence between the set of all g-essential ideals of N and the set of all generalised f -essential full ideals of M n (N ). Proof.
2. Let T be a f -superfluous full ideal of M n (N ) such that I ⋆ ∩ T = (0).Since T is a full ideal, we have T = K ⋆ for some ideal K of N .Therefore, by Lemma 3.2 we get (I ∩ K) ⋆ = I ⋆ ∩ K ⋆ = (0) and so I ∩ K = (0).Since T ≪ f M n (N ), by Lemma 3.13 (2), we have A is a full ideal }.Define the mappings φ : P → Q by φ(A) = A ⋆ and ψ : Q → P by ψ(A) = A ⋆ .Then the correspondence follows, similar to the Lemma 3.9(3).

Generalised essential ideal graph
In this section, we introduce the notion of generalized essential ideal graph (in short, g-essential ideal graph) of a module H over a nearring N (denoted as, L g (H)).We derive properties such as diameter, completeness based on a given N -group.

Definition 4.1. The g-essential ideal graph, L g (H) of H, is a graph whose vertex set is the set of all non-trivial ideals of H, and two distinct vertices A, B are adjacent if
, and H = N .The non-trivial ideals of H are t , t + 1 , t 2 + t and t 2 + 1 .The ideal t 2 + t is superfluous.Since t 2 + 1 ∩ t 2 + t = (0), we have t 2 + 1 is non-g-essential.The corresponding g-essential ideal graph is given in the Figure 1.

If H has a unique non-zero superfluous ideal, say B, then B is a universal vertex in L g (H).
We denote M ax(H) = {M : M is a maximal ideal of H}.Theorem 4.7.[23] Let N = N 0 and H be finitely generated.Then every proper ideal of H is contained in a maximal ideal.In particular, H has a maximal ideal.Proposition 4.8.Let H be a completely reducible N -group.Then M .On the contrary, suppose that B is non-zero.Since H is completely reducible, we have B is a direct summand.Therefore, there exists a proper ideal S of H such that B + S = H.As S = H and H is finitely generated, we get S ⊆ X for some X ∈ M ax(H).Now since B =   ⋆ 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 2 2 5 0 0 0 0 0 0 2 2 6 0 0 0 0 0 0 2 2 7 0 0 0 0 0 0 2 2  B ∈ V (P g (H)).Since every non-maximal ideal is a minimal ideal, by Corollary 4.17, we get that L g (H) is a complete graph, which implies A ∼ B ∈ E(L g (H)).Since P g (H) is an induced subgraph of L g (H), we get A ∼ B ∈ E(P g (H)), a contradiction.Therefore, P g (H) is an empty graph.Proposition 4.30.There exists a path between any two superfluous ideals in P g (H).
Proof.Let I, J be two distinct superfluous ideals of H such that I, J ∈ V (P g (H)).If I + J ≤ ge H, then I ∼ J ∈ E(P g (H)).Assume that I + J is not g-essential.Let S I , S J be g-complements of I and J respectively.Since I and J are superfluous ideals, we have that S I and S J are not g-essential.Therefore, S I , S J ∈ V (P g (H)).Also, by Lemma 4.21, we get I + S I ≤ ge H and J + S J ≤ ge H.If I + S J or J + S I is g-essential, then either I ∼ S J ∼ J or I ∼ S J ∼ J is an IJ path.If neither J + S I nor I + S J is g-essential, then J + S I , I + S J ∈ V (P g (H)) and hence I ∼ J + S I ∼ I + S J ∼ J is a path from I to J.

Conclusion
We have defined g-essential ideal graph of an N -group.For a finitely generated N -group H, we have shown that the maximal ideal is always a universal vertex and hence the g-essential ideal graph of such N -groups is always connected with diameter not more than 2. We have obtained several properties of g-essential ideal graphs based on the notions of connectivity, completeness etc.As future scope, we will explore to study the lattice aspects and graph theoretical properties of essential elements and superfluous elements as motivated by the authors in [26].Furthermore, the notions discussed in this paper can be extended to study the finite dimensional aspects in matrix nearrings.

Definition 3 . 8 .
S M n (N ) is called an f -essential ideal if for any full ideal T of M n (N ), S ∩ T = (0) implies T = (0)and it is denoted by S ≤ f e M n (N ).Proposition 3.9.Let I N and S M n (N ).Then 1. S ≤ f e M n (N ) implies S ⋆ ≤ e N .2. I ≤ e N implies I ⋆ ≤ f e M n (N ).

Proposition 4 . 22 .
D N H such that (C + K) + D = H.Since C + (K + D) = H and C ≪ H, we have K + D = H.Again, since K is superfluous in H, we get D = H.Therefore, C + K ≪ H. Since C is a g-complement of A and C + K ≪ H satisfying A ∩ (C + K) = (0), we get a contradiction.Therefore, A + C ≤ ge H. Every non g-essential ideal of H is adjacent to its g-complement in L g (H).
9.If S ≤ ge H, then S + P ≤ ge H for any ideal P of H.Proof.Let S ≤ ge H and P N H.To prove S + P is g-essential, let K ≪ H such that (S + P ) ∩ K = (0).Now S ∩ K ⊆ (S + P ) ∩ K = (0), implies S ∩ K = (0).Since S ≤ ge H, we get K = (0).Therefore,S + P ≤ ge H. Any proper g-essential ideal of H is a universal vertex in L g (H).Proof.Let A N H which is proper and g-essential.Let B be a non-trivial ideal of H.We prove A ∼ B ∈ E(L g (H)).Since A ≤ ge H, by Proposition 4.9, we have A+B ≤ ge H and soA ∼ B ∈ E(L g (H)).Since B is arbitrary, A ∼ B ∈ E(L g (H)) for all B ∈ V (L g (H)), we conclude that A is a universal vertex in L g (H).H, which means B is not superfluous.Hence H has no non-zero superfluous ideals and every proper ideal of H is g-essential.By Lemma 4.10, we have every proper g-essential ideal is a universal vertex and hence L g (H) is a complete graph.getB∼K∈E(Lg (H)), a contradiction, as L g (H) is empty.Conversely, if H has exactly one non-zero, non g-essential ideal, then L g (H) = K 1 , an empty graph.Assume that every non-zero, non-g-essential ideal of M is a minimal ideal.To prove L g (H) is a complete graph, let A and B be two distinct vertices of L g (H).Then A and B are two non-zero proper ideals of H. Case 1: Either A or B is g-essential.In this case, by Proposition 4.10, A ∼ B ∈ E(L g (H)).Case 2: Neither A nor B is g-essential.SinceA and B are non-zero, from the hypothesis A and B are minimal.Then (0) = A A + B, implies that A + B is not minimal.Again from the hypothesis, A + B ≤ ge H, and hence A ∼ B ∈ E(L g (H)).Conversely, suppose that L g (H) is a complete graph.We prove every non-zero non-g-essential ideal is a minimal ideal.On the contrary, let (0) = A ge H which is not minimal.Then, there exists(0) = B N H such that B A. Since L g (H) is complete, A ∼ B ∈ E(L g (H)), which implies A + B = A ≤ ge H, a contradiction to the assumption.Thus, every non-zero, non-g-essential ideal is a minimal ideal.If every non-maximal ideal of H is a minimal ideal, then L g (H) is a complete graph.Proof.Follows from Proposition 4.13 and 4.16.Let A N H.An ideal B of H is a g-complement of A in H if B is maximal with respect to (A ∩ B = (0)and B is superfluous in H).Example 4.19.Consider the nearring N = (Z 2 × Z 2 × Z 2 , +, •) where the addition is carried out component-wise modulo 2 and the multiplication table is in the Table ??.Elements are denoted as given in the Example 4.
non-zero ideal of H. Since K B = (0), we have K L for some L ∈ M ax(H).Now L B + L ⊆ H and L ∈ M ax(H) imply B + L =

Table 3 :
The multiplication table of Z 2 × Z 2 × Z 2 I 6 = {0, 1} and I 7 = {0}.The g-complement of I 6 is I 5 .Let A N H and C be its g-complement.Then A + C ≤ ge H.