Total Zero-Divisor Graph of A Field

Total Zero-Divisor Graph of A Field

Nandhini. R, Akshaya. G, Kayalvizhi. M Thassim Beevi Abdul Kader College for Women, Kilakarai

Abstract: Let F be a Field with z(F), its set of zero divisors. The total zero divisor graph of F, denoted Z((F)) is the undirected (simple) graph with vertices Z()=Z(F)-{0}, the set of non- zero, zero divisors of F, and for distinct x, y () the vertices x and y are adjacent if and only if x+ y z(F). In this paper, we study if Z((F)) is finite and every vertex of Z((F)) has a finite degree then F is finite and also prove that Z((F)) connected with diam3.

1. INTRODUCTION

   In this paper, we study the total zero divisor graph is the (undirected) graph with vertices Z(F)=Z(F)- {0}. The set of non-zero zero divisor of F and for distinct x, y z(F), the vertices x and y are adjacent if and only if x+ y z(F). It is denoted by Z((F)) and is the (induced) subgraph of total graph. We show that Z((F)) is finite then F is finite and not an integral domain, if every vertex of Z((F)) has finite degree then F is finite and also prove that Z((F)) is connected with diam3. For some other recent papers on zero divisor graphs.

   1. Definition:

      A ring R is called a coloring if (0()) is finite.

   2. Definition:

      An element xR is said to be a zero divisor if there exists some element 0yR such that xy=0.

   3. Presumption:

      (0()) = 1 if and only if R = {0}.

   4. presumption:

      (0()) = 2 if and only if R is an integral domain, R 4

      or R 2[]()2.

   5. Definition:

   The chromatic number of a zero-divisor graph of a ring R is equal to the clique number of the ring. That is, (0())

   =cl(R).

   III. MAIN RESULT

   2.1Definition:

2. PRELIMINARIES

   3.1Theorem:

   Let F be field then the total zero divisor graph if finite

   The number of edges incident with a vertex Vis called the degree of V and it is denoted by d(V). The minimum and maximum degree of a vertex of a graph are respectively denoted by and .

   1. Definition:

      A graph G in which every vertex is adjacent to every other vertex is called a complete graph. Complete graph is represented as where n is the number of vertices in

   2. Definition:

      The chromatic number of a zero-divisor graph of a ring R, denoted by (0()) is the minimal number of colors required to assign each vertex in a zero-divisor graph a color so that no two adjacent vertices are assigned the same color.

   3. Definiton:

      A graph 0() is a k-colorable if 0() can be colored with less than or equal to k colors.

   4. Defnition:

      A graph G is said to be a connected graph. If there is at least one path between every pair of vertices in G. otherwise G is said to be a disconnected graph.

   5. Definition:

   if and only if either or an integral domain. In particular if 1 Z((F)) .Then F ia finite.

   Proof;

   Let F be a field and Z(F) be the set of zero divisors in F and Let Z((F)) be the total zero divisor graph. Then all vertices of Z((F)) is non-zero, zero divisor of F.

   It is trivial that if F is finite then Z((F)) is also finite.

   Suppose that Z((F) is finite and non-empty. This implies that Z(F) is finite, suppose these are two elements u, vF, u0,v0.suchthat u+vZ(F)

   Let I=Ann(Z), then u+vI

   Since u+v ()this implies that I () further I is finite and f(u+v) I for all fF.[u+v ,fFf(u+v) I]suppose F is finite.Then there is an iI suchthat K={fF/f(u+v)=i}is infinite.

   For any f, tK

   F(u+v) = i, t(u+v) = i (ft) (u+v) =0

   (ft) Ann(u+v) {since, KAnn(u+v), K is ftJf-tVAnn(u+v) infinite)

   Where kAnn(u+v) Since f-tZ(F)

   i.e. Ann(u+v) Z(F), is infinite, a contradiction therefore F must be finite.

   1. Theorem:

      Let f be a field with identity. Then S = F × 0 is a field

      Any two distinct vertices a and b in graph G, the distance between a and b, denoted by d (a, b) is the length of a shortest path connecting a and b.

      without identity, S = Z(S), and

      (S)

      2

      (F).

      Proof: Clearly S = Z(S) and T has no identity. Define : F/

      S / by ([u]) = [(u, 0)]. It is easily verified that (u)

      = (v) for u, v F if and only if ((u, 0)) = ((v, 0)), and [(u, 0)] = [(u, 1)] for every u F. Thus, is a well-defined bijection. Moreover, restricts to a graph isomorphism from (F) to (S) since [(u, 0)] [(v, 0)] = [(0, 0)] if and only if [u][v] = [0].

   2. Theorem:

      Let F be a field such that Z(F) Is not an ideal of F then Z(F) is connected with diamZ(F)=2

      Proof: Each u () is adjacent to 0. Thus, u0v is a path in Z(F) of length two between any two distinct u, v () Moreover, there are non-adjacent u, v () since Z(F) is not an ideal of F.

      So, diamZ(F)=2.

      Hence proved

   3. Theorem: Let F be a field then Z(F) is connected with diam3

   Proof:

   Let u, v be vertices in Z((F)),

   There exists u+z Z(F), v+WZ(F)

   If u+v Z(F) then uv is a path of length is perpendicular containing u, v.

   If u+v Z(F) and w+z Z(F) then u and v are contained by a path uwv of lengtp

   If u+v Z(F) and w+z Z(F) then u and v are connected by a path uv of length=2.

   Hence proved.

   IV. REFERENCE

   1. I. Beck, coloring of commutative ring, J. Algebra, 116(1) (1988),208-226.

   2. D.F. Anderson and P.S. Livingston, the zero-divisor graph of commutative ring, J. Algebra, 217(2) (1999), 434-447

   3. A.R. Ashrafi and A. Tadayyonfar, The zero-divisor graph of 2×2 matrices over a field, vol.39., (2016), 977-990