On Sum Element Ideal Graphs

fryad abdulqadr

Abstract


  Let R be a commutative ring with identity and let x be an element of R. The element ideal graph is a graph whose vertex set is the set of non-trivial ideals of R and two distinct vertices I and J are adjacent if and only if xIJ. In this paper, we consider a new kind of graph  associated with R denoted by  whose vertex set is the set of non-trivial ideals of R and two distinct vertices I and J are adjacent if and only if I+J is a vertex of .


References


REFERENCES

Aalipour, G., Akbari, S., Nikandish, R., coloring of the annihilating ideal graph of a commutative ring, Journal of Discrete Math., 312 , 2620-2625.

Abdul-Qadr, F. H. and Shuker, N. H. 2015, Annihilating and element ideal graphs of commutative rings, Ph.D-Dissertation, University of Mosul.

Akbari, S., Kami, D., Mohammadi, F. and Moradi, S. 2009, The total graph and regular graph of a commutative ring, Journal of Pure and Applied Algebra, 213(2), 2224 - 2228.

Akbari, S. , Maimani, H. R. and Yassemim S. 2003, when zero divisor graph is planner or a complete r-partite graph, J. Algebra, 270(1), 169-180.

Anderson, D. F. and Badawi, A. The generalized total graph of a commutative ring, Journal of Algebra and Appl., 12(5), (2013), Art. ID: 1250212, (18 pages).

Anderson, D. F. and Livingston, P. S. 1999 , The zero divisor graph of a commutative ring, Journal of Algebra, 217, 434-447.

Anderson, D. D. and Naseer, M. 1993, Beckʼs coloring of a commutative ring, Journal of Algebra, 159, 500-514.

Badawi, A. 2014, On the annihilator graph of a commutative ring, Comm. in Algebra, 42(1), 108-121.

Beck, I. 1988, Coloring of commutative ring, Journal of Algebra 116(1), 206-226.

Behboodiand, M. and Rakeei, Z. 2011, The Annihilating-Ideal Graph of Commutative Rings I, Journal of Algebra and Appl., 10, 727–739.

Behboodiand, M. and Rakeei, Z. 2011, The Annihilating-Ideal Graph of Commutative Rings I, Journal of Algebra and Appl., 10, 741–753.

Burton, D. M.1970, A First Course in Rings and Ideals, Addison Wesley Publishing Company, Inc.

Chartrand, G. and Lesniak, L. 1986, Graphs and Digraphs, 2nd ed., Wadsworth and Brooks/Cole, California.

Levy, R. and Shapiro, J. 2002, The zero divisor graph of nonneam regular rings, Journal of comm. Algebra 30(2), 745-750.

Smith, N. O. 2003, Planar zero-divisor graphs, International Journal of Commut. Rings, 2(4), 177-188.




Copyright (c) 2017 ZANCO Journal of Pure and Applied Sciences

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

ZANCO Journal of Pure and Applied Sciences (print version: ISSN 2218-0230 online version: ISSN 2412-3986, DOI: 10.21271) is published by Salahaddin University-Erbil / Department of Scientific Publications. Responsibility for the contents rests upon the authors and not upon  Salahaddin University-Erbil or the Journal Editorial and Advisory Boards. 

Department of Scientific Publication Office: The Central Library of Salahaddin University-Erbil, Kirkuk Road, Erbil, Kurdistan, Iraq. Cell Phone: +964 (0)750 7761675, email: zanco.scientific@su.edu.krd. www.su.edu.krd, www.zancojournals.su.edu.krd

Copyright and Reprint Permission: It is the policy of ZANCO to own the copyright to the technical contributions it publishes and to facilitate the appropriate reuse of this material by others. Photocopying is permitted with credit to the source for individuals for individual use.

Copyright © 2017. All Rights Reserved. Salahaddin University-Erbil