A Refinement of Cayley Graphs Associated to Rings A Refinement of Cayley Graphs Associated to A. R. Naghipour Rings Shahrekord University, Iran. Introduction A. R. Naghipour A Refinement Shahrekord University, Iran. of Cayley Graphs Associated to Rings 5 March 2018 Discrete Maths Research Group. 1 / 29
Outline A Refinement of Cayley Graphs Associated to Rings A. R. Naghipour 1 Introduction Shahrekord University, Iran. 2 A Refinement of Cayley Graphs Associated to Rings Introduction A Refinement of Cayley Graphs Associated to Rings 2 / 29
Introduction A Refinement of Cayley Some notations and structure for commutative rings Graphs Associated to Rings A. R. Examples of commutative rings Naghipour Shahrekord University, (1) Z n = Z /n Z . Iran. (2) F n = field of order n . Introduction A Refinement Ideals and Maximal ideals of Cayley Graphs Associated to Let R be a commutative Ring with identity. Rings (1) An ideal in R is an additive subgroup I ⊆ R such that Ix ⊆ I for all x ∈ R . (2) I is called maximal ideal if there is no ideal J with I � J � R . 3 / 29
Introduction A Refinement of Cayley Some notations and structure for commutative rings Graphs Associated to Rings A. R. Examples of commutative rings Naghipour Shahrekord University, (1) Z n = Z /n Z . Iran. (2) F n = field of order n . Introduction A Refinement Ideals and Maximal ideals of Cayley Graphs Associated to Let R be a commutative Ring with identity. Rings (1) An ideal in R is an additive subgroup I ⊆ R such that Ix ⊆ I for all x ∈ R . (2) I is called maximal ideal if there is no ideal J with I � J � R . 3 / 29
Introduction A Refinement of Cayley Some notations and structure for commutative rings Graphs Associated to Rings A. R. Examples of commutative rings Naghipour Shahrekord University, (1) Z n = Z /n Z . Iran. (2) F n = field of order n . Introduction A Refinement Ideals and Maximal ideals of Cayley Graphs Associated to Let R be a commutative Ring with identity. Rings (1) An ideal in R is an additive subgroup I ⊆ R such that Ix ⊆ I for all x ∈ R . (2) I is called maximal ideal if there is no ideal J with I � J � R . 3 / 29
Local rings A Refinement of Cayley Graphs Associated to Rings Definition of local ring A. R. Naghipour Call a ring R local if R has exactly one maximal ideal. Shahrekord University, Iran. Examples of local rings Introduction (1) Z 4 . A Refinement of Cayley (2) Z 9 . Graphs Associated to (3) Z p 2 , where p is a prime number. Rings (4) Z p [ X ] ( X 2 ) , where p is a prime number. 4 / 29
Local rings ( Z 4 and Z 9 ) A Refinement of Cayley Graphs Associated to Rings 1 A. R. 1,3 Naghipour 8 Shahrekord 2 University, Iran. 0,3,6 0,2 Introduction 7 4 A Refinement of Cayley 5 Graphs Associated to Rings {0,1,2,...,8} {0,1,2,3} 9 4 5 / 29
Structure of finite commutative rings A Refinement (1) of Cayley Graphs Let R be a finite commutative ring. Then Associated to Rings A. R. R = R 1 × R 2 × · · · × R k , Naghipour Shahrekord University, where R i is a local ring. Iran. Introduction (2) A Refinement of Cayley Let R be a finite commutative ring. Then Graphs Associated to Rings R/J ( R ) = F 1 × F 2 × · · · × F k , where F i is a Field. (Here J ( R ) , the Jacobson radical of R , is the intersection of all maximal ideals of R ) 6 / 29
Some facts about Jacobson radical A Refinement of Cayley Graphs Associated to Rings A. R. Naghipour Structure of Jacobson radical Shahrekord University, J ( R ) = { r ∈ R | 1 + rx is unit for all x ∈ R } . Iran. Introduction Theorem A Refinement of Cayley Graphs u is a unit in R if and only if u + J ( R ) is a unit in R/J ( R ) . Associated to Rings 7 / 29
A Refinement Some important graphs associated to rings of Cayley Graphs (1) Zero divisor Graph of a ring. Associated to Rings (2) Cayley Garph of a ring. A. R. (3) Unit Graph of a ring. Naghipour Shahrekord University, Iran. (1) Zero divisor graph of a ring Introduction The concept of a zero-divisor graph of a commutative ring was A Refinement first introduced by Beck. (In his work all elements of the ring of Cayley Graphs were vertices of the graph). Associated to Rings V (Γ( R ) = Z ( R ) \ { 0 } and two distinct vertices x and y are adjacent if and and only if xy = 0 . [Beck] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208-226. 8 / 29
Some examples of Zero divisor graphs A Refinement of Cayley Graphs Associated to Rings A. R. Naghipour [ ] x [ ] x [ ] x Shahrekord 2 3 2 or ( , or ( , or ( University, 4 9 2 2 6 8 2 2 3 ) ) ) x x x Iran. Introduction A Refinement of Cayley Graphs Associated to Rings [ , ] x y [ ] x [ ] x 2 4 5 or or ( 3 3 25 2 2 2 2 ( , , ) ( ) x ) x xy y x 9 / 29
Some examples of Zero divisor Graphs A Refinement of Cayley Graphs Associated to Rings A. R. Naghipour Shahrekord University, Iran. Introduction A Refinement of Cayley Graphs Associated to Rings 2 4 2 7 3 5 10 / 29
A Refinement (2) Cayley graph of a ring of Cayley Graphs The Cayley graph Γ( R ) is the graph with vertex set R such Associated to Rings that two distinct vertices x and y are adjacent if and only if A. R. x − y is unit in R . Unitary Cayley graphs are introduced in: Naghipour Shahrekord University, Iran. Lucchini, et al. Introduction A. Lucchini, A. Maroti, Some results and questions related to A Refinement the generating graph of a finite group, Proceedings of the of Cayley Graphs Ischia Group Theory Conference, 2008. Associated to Rings Akhtar, et al. R. Akhtar, M. Boggess, T. Jackson-Henderson, I. Jim´enez, R. Karpman, A. Kinzel and D. Pritikin, On the unitary Cayley graph of a finite ring, Electron. J. Combin. 16 (2009) # R117. 11 / 29
A general example of Cayley graph A Refinement of Cayley ( R, M ) is a local ring, R/M = { x 1 + M, x 2 + M, . . . , /x t + M } Graphs Associated to and | x i + M | = n i | for all 1 ≤ i ≤ t . Rings A. R. Naghipour Shahrekord x M 1 University, Iran. K n 1 Introduction A Refinement x M x M K K of Cayley n 2 n n t 2 t Graphs Associated to Rings K K n n 3 i x M x M n 3 i 12 / 29
A Refinement of Cayley Graphs Associated to Rings (3) Unit graph of a ring A. R. Naghipour The unit graph Γ( R ) is the graph with vertex set R such that Shahrekord University, two distinct vertices x and y are adjacent if and only if x + y is Iran. unit in R . The unit graphs are introduced in: Introduction A Refinement of Cayley Fuchs Graphs Associated to E. Fuchs, Longest induced cycles in circulant graphs, Electron. Rings J. Combin. 14 (2005) # R52. 13 / 29
Some examples unit graphs A Refinement of Cayley Graphs Associated to Rings A. R. Naghipour Shahrekord University, Iran. 3 4 2 2 Introduction A Refinement of Cayley Graphs Associated to Rings 6 2 3 3 3 14 / 29
A Refinement of Cayley Graphs Associated to Rings Definition in this talk A. R. A Refinement of Cayley Graphs Associated to Rings Naghipour Shahrekord Let R be a finite ring and U ( R ) be the set of all unit elements University, Iran. of R . The Unit graph Γ( R ) is the graph with vertex set R such Introduction that two distinct vertices x and y are adjacent if and only if A Refinement there exists a unit element u of R such that x + uy is unit in R . of Cayley Graphs Associated to Rings If we omit the word ”distinct”, we obtain the graph Γ ℓ ( R ) ; this graph may have loops. 15 / 29
A Refinement of Cayley Graphs Associated to Rings Motivation A. R. Naghipour (1) The study of algebraic structures using the properties of Shahrekord University, graphs, Iran. (2) Some result about unit 1-stable range rings. Introduction A Refinement of Cayley We recall that a ring R is said to have unit 1-stable range if, Graphs Associated to whenever Rx + Ry = R , there exists u ∈ U ( R ) such that Rings x + uy ∈ U ( R ) . 16 / 29
Some examples A Refinement of Cayley Graphs 0 0 1 Associated to 0 Rings A. R. Naghipour 1 2 1 Shahrekord 3 2 University, ) ) ) ) ) Iran. ( ( ( ( ( 2 2 3 4 4 0 Introduction 8 1 A Refinement (0,1) (0,0) (0,2) of Cayley 2 7 Graphs Associated to Rings 6 3 (1,2) (1,0) 5 4 (1,1) ( ) ( ) ( ) 2 3 2 3 9 17 / 29
Outline A Refinement of Cayley Graphs Associated to Rings A. R. Naghipour 1 Introduction Shahrekord University, Iran. 2 A Refinement of Cayley Graphs Associated to Rings Introduction A Refinement of Cayley Graphs Associated to Rings 18 / 29
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