An ideal-based zero-divisor graph of direct products of commutative rings

S. Ebrahimi Atani; M. Shajari Kohan; Z. Ebrahimi Sarvandi

Discussiones Mathematicae - General Algebra and Applications (2014)

  • Volume: 34, Issue: 1, page 45-53
  • ISSN: 1509-9415

Abstract

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In this paper, specifically, we look at the preservation of the diameter and girth of the zero-divisor graph with respect to an ideal of a commutative ring when extending to a finite direct product of commutative rings.

How to cite

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S. Ebrahimi Atani, M. Shajari Kohan, and Z. Ebrahimi Sarvandi. "An ideal-based zero-divisor graph of direct products of commutative rings." Discussiones Mathematicae - General Algebra and Applications 34.1 (2014): 45-53. <http://eudml.org/doc/270743>.

@article{S2014,
abstract = {In this paper, specifically, we look at the preservation of the diameter and girth of the zero-divisor graph with respect to an ideal of a commutative ring when extending to a finite direct product of commutative rings.},
author = {S. Ebrahimi Atani, M. Shajari Kohan, Z. Ebrahimi Sarvandi},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {zero-divisor graph; ideal-based; diameter; girth; finite direct product; commutative semirings; -ideals; -primary ideals; zero-divisors; -semidomains; -semidomainlike semirings},
language = {eng},
number = {1},
pages = {45-53},
title = {An ideal-based zero-divisor graph of direct products of commutative rings},
url = {http://eudml.org/doc/270743},
volume = {34},
year = {2014},
}

TY - JOUR
AU - S. Ebrahimi Atani
AU - M. Shajari Kohan
AU - Z. Ebrahimi Sarvandi
TI - An ideal-based zero-divisor graph of direct products of commutative rings
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2014
VL - 34
IS - 1
SP - 45
EP - 53
AB - In this paper, specifically, we look at the preservation of the diameter and girth of the zero-divisor graph with respect to an ideal of a commutative ring when extending to a finite direct product of commutative rings.
LA - eng
KW - zero-divisor graph; ideal-based; diameter; girth; finite direct product; commutative semirings; -ideals; -primary ideals; zero-divisors; -semidomains; -semidomainlike semirings
UR - http://eudml.org/doc/270743
ER -

References

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  1. [1] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra. 217 (1999) 434-447. doi: 10.1006/jabr.1998.7840. Zbl0941.05062
  2. [2] M. Axtell, J. Stickles and J. Warfel, Zero-divisor graphs of direct products of commutative rings, Houston J. Math. 32 (2006) 985-994. Zbl1110.13004
  3. [3] D.F. Anderson, M.C. Axtell and J.A. Stickles Jr., Zero-divisor graphs in commutative rings in commutative Algebra-Noetherian and Non-Noetherian Perspectives (M. Fontana, S.E. Kabbaj, B. Olberding, I. Swanson, Eds), (Springer-Verlag, New York, 2011) 23-45. doi: 10.1007/978-1-4419-6990-3_2 
  4. [4] D.F. Anderson and A. Badawi, On the zero-divisor graph of a ring, Comm. Algebra 36 (2008) 3073-3092. doi: 10.1080/00927870802110888. Zbl1152.13001
  5. [5] I. Beck, Coloring of commutative rings, J. Algebra. 116 (1998) 208-226. doi: 10.1016/0021-8693(88)90202-5. 
  6. [6] S. Ebrahimi Atani and M. Shajari Kohan, On L-ideal-based L-zero-divisor graphs, Discuss. Math. Gen. Algebra Appl. 31 (2011) 127-145. doi: 10.7151/dmgaa.1178. Zbl1255.05094
  7. [7] S. Ebrahimi Atani and M. Shajari Kohan, L-zero-divisor graphs of direct products of L-commutative rings, Discuss. Math. Gen. Algebra Appl. 31 (2011) 159-174. doi: 10.7151/dmgaa.1180. Zbl1255.05095
  8. [8] S.P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Algebra 31 (2003) 4425-4443. doi: 10.1081/AGB-120022801. Zbl1020.13001

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