Research Article
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Year 2018, Volume 23, Issue 23, 115 - 130, 11.01.2018
https://doi.org/10.24330/ieja.373650

Abstract

References

  • F. Aliniaeifard, M. Behboodi, E. Mehdi-Nezhad and A. M. Rahimi, The annihilating-ideal graph of a commutative ring with respect to an ideal, Comm. Algebra, 42(5) (2014), 2269-2284.
  • D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra, 159(2) (1993), 500-514.
  • D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
  • D. F. Anderson, R. Levy and J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra, 180(3) (2003), 221-241.
  • D. F. Anderson, A. Frazier, A. Lauve and P. S. Livingston, The zero-divisor graph of a commutative ring, II, in Ideal Theoretic Methods in Commutative Algebra, Lecture Notes in Pure and Appl. Math., 220, Dekker, New York, (2001), 61-72.
  • D. F. Anderson, S. Ghalandarzadeh, S. Shirinkam and P. Malakooti Rad, On the diameter of the graph 􀀀Ann(M)(R), Filomat, 26(3) (2012), 623-629.
  • H. Ansari-Toroghy and F. Farshadifar, The dual notion of multiplication mod- ules, Taiwanese J. Math., 11(4) (2007), 1189-1201.
  • H. Ansari-Toroghy and F. Farshadifar, Strong comultiplication modules, CMU. J. Nat. Sci., 8(1) (2009), 105-113.
  • H. Ansari-Toroghy and F. Farshadifar, The dual notions of some generaliza- tions of prime submodules, Comm. Algebra, 39(7) (2011), 2396-2416.
  • H. Ansari-Toroghy and F. Farshadifar, On the dual notion of prime submod- ules, Algebra Colloq., 19(spec 1) (2012), 1109-1116.
  • H. Ansari-Toroghy and F. Farshadifar, Fully idempotent and coidempotent modules, Bull. Iranian Math. Soc., 38(4) (2012), 987-1005.
  • H. Ansari-Toroghy and F. Farshadifar, On the dual notion of prime radicals of submodules, Asian-Eur. J. Math., 6(2) (2013), 1350024 (11 pp).
  • I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
  • G. A. Cannon, K. M. Neuerburg and S. P. Redmond, Zero-divisor graphs of nearrings and semigroups, in Nearrings and Near elds (eds: H. Kiechle, A. Kreuzer, M.J. Thomsen), Springer, Dordrecht, (2005), 189-200.
  • S. Ceken, M. Alkan and P. F. Smith, The dual notion of the prime radical of a module, J. Algebra, 392 (2013), 265-275.
  • P. Dheena and B. Elavarasan, An ideal-based zero-divisor graph of 2-primal near-rings, Bull. Korean Math. Soc., 46(6) (2009), 1051-1060.
  • S. Ebrahimi Atani and A. Youse an Darani, Zero-divisor graphs with respect to primal and weakly primal ideals, J. Korean Math. Soc., 46(2) (2009), 313-325.
  • L. Fuchs, W. Heinzer and B. Olberding, Commutative ideal theory without niteness conditions: Irreducibility in the quotient led, in: Abelian Groups, Rings, Modules and Homological Algebra, Lect. Notes Pure Appl. Math., 249, Chapman & Hall/CRC, Boca Raton, FL, (2006), 121-145.
  • Sh. Ghalandarzadeh, S. Shirinkam and P. Malakooti Rad, Annihilator ideal- based zero-divisor graphs over multiplication modules, Comm. Algebra, 41(3) (2013), 1134-1148.
  • I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.
  • H. R. Maimani, M. R. Pournaki and S. Yassemi, Zero-divisor graph with respect to an ideal, Comm. Algebra, 34(3) (2006), 923-929.
  • S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Algebra, 31(9) (2003), 4425-4443.
  • S. Yassemi, Maximal elements of support and cosupport, May 1997, http://streaming.ictp.it/preprints/P/97/051.pdf.
  • S. Yassemi, The dual notion of prime submodules, Arch. Math. (Brno), 37(4) (2001), 273-278.
  • A. Youse an Darani, Notes on the ideal-based zero-divisor graph, J. Math. Appl., 32 (2010), 103-107.

On the ideal-based zero-divisor graphs

Year 2018, Volume 23, Issue 23, 115 - 130, 11.01.2018
https://doi.org/10.24330/ieja.373650

Abstract

Let R be a commutative ring. In this paper, we study the annihilator
ideal-based zero-divisor graph by replacing the ideal I of R with the ideal
AnnR(M) for an R-module M. Also, we investigate a certain subgraph of the
annihilator ideal-based zero-divisor graph and obtain some related results.

References

  • F. Aliniaeifard, M. Behboodi, E. Mehdi-Nezhad and A. M. Rahimi, The annihilating-ideal graph of a commutative ring with respect to an ideal, Comm. Algebra, 42(5) (2014), 2269-2284.
  • D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra, 159(2) (1993), 500-514.
  • D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
  • D. F. Anderson, R. Levy and J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra, 180(3) (2003), 221-241.
  • D. F. Anderson, A. Frazier, A. Lauve and P. S. Livingston, The zero-divisor graph of a commutative ring, II, in Ideal Theoretic Methods in Commutative Algebra, Lecture Notes in Pure and Appl. Math., 220, Dekker, New York, (2001), 61-72.
  • D. F. Anderson, S. Ghalandarzadeh, S. Shirinkam and P. Malakooti Rad, On the diameter of the graph 􀀀Ann(M)(R), Filomat, 26(3) (2012), 623-629.
  • H. Ansari-Toroghy and F. Farshadifar, The dual notion of multiplication mod- ules, Taiwanese J. Math., 11(4) (2007), 1189-1201.
  • H. Ansari-Toroghy and F. Farshadifar, Strong comultiplication modules, CMU. J. Nat. Sci., 8(1) (2009), 105-113.
  • H. Ansari-Toroghy and F. Farshadifar, The dual notions of some generaliza- tions of prime submodules, Comm. Algebra, 39(7) (2011), 2396-2416.
  • H. Ansari-Toroghy and F. Farshadifar, On the dual notion of prime submod- ules, Algebra Colloq., 19(spec 1) (2012), 1109-1116.
  • H. Ansari-Toroghy and F. Farshadifar, Fully idempotent and coidempotent modules, Bull. Iranian Math. Soc., 38(4) (2012), 987-1005.
  • H. Ansari-Toroghy and F. Farshadifar, On the dual notion of prime radicals of submodules, Asian-Eur. J. Math., 6(2) (2013), 1350024 (11 pp).
  • I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
  • G. A. Cannon, K. M. Neuerburg and S. P. Redmond, Zero-divisor graphs of nearrings and semigroups, in Nearrings and Near elds (eds: H. Kiechle, A. Kreuzer, M.J. Thomsen), Springer, Dordrecht, (2005), 189-200.
  • S. Ceken, M. Alkan and P. F. Smith, The dual notion of the prime radical of a module, J. Algebra, 392 (2013), 265-275.
  • P. Dheena and B. Elavarasan, An ideal-based zero-divisor graph of 2-primal near-rings, Bull. Korean Math. Soc., 46(6) (2009), 1051-1060.
  • S. Ebrahimi Atani and A. Youse an Darani, Zero-divisor graphs with respect to primal and weakly primal ideals, J. Korean Math. Soc., 46(2) (2009), 313-325.
  • L. Fuchs, W. Heinzer and B. Olberding, Commutative ideal theory without niteness conditions: Irreducibility in the quotient led, in: Abelian Groups, Rings, Modules and Homological Algebra, Lect. Notes Pure Appl. Math., 249, Chapman & Hall/CRC, Boca Raton, FL, (2006), 121-145.
  • Sh. Ghalandarzadeh, S. Shirinkam and P. Malakooti Rad, Annihilator ideal- based zero-divisor graphs over multiplication modules, Comm. Algebra, 41(3) (2013), 1134-1148.
  • I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.
  • H. R. Maimani, M. R. Pournaki and S. Yassemi, Zero-divisor graph with respect to an ideal, Comm. Algebra, 34(3) (2006), 923-929.
  • S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Algebra, 31(9) (2003), 4425-4443.
  • S. Yassemi, Maximal elements of support and cosupport, May 1997, http://streaming.ictp.it/preprints/P/97/051.pdf.
  • S. Yassemi, The dual notion of prime submodules, Arch. Math. (Brno), 37(4) (2001), 273-278.
  • A. Youse an Darani, Notes on the ideal-based zero-divisor graph, J. Math. Appl., 32 (2010), 103-107.

Details

Journal Section Articles
Authors

Habibollah ANSARİ-TOROGHY This is me


Faranak FARSHADİFAR>

0000-0001-7600-994X


Farideh MAHBOOBİ-ABKENAR This is me

Publication Date January 11, 2018
Published in Issue Year 2018, Volume 23, Issue 23

Cite

Bibtex @research article { ieja373650, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2018}, volume = {23}, number = {23}, pages = {115 - 130}, doi = {10.24330/ieja.373650}, title = {On the ideal-based zero-divisor graphs}, key = {cite}, author = {Ansari-toroghy, Habibollah and Farshadifar, Faranak and Mahboobi-abkenar, Farideh} }
APA Ansari-toroghy, H. , Farshadifar, F. & Mahboobi-abkenar, F. (2018). On the ideal-based zero-divisor graphs . International Electronic Journal of Algebra , 23 (23) , 115-130 . DOI: 10.24330/ieja.373650
MLA Ansari-toroghy, H. , Farshadifar, F. , Mahboobi-abkenar, F. "On the ideal-based zero-divisor graphs" . International Electronic Journal of Algebra 23 (2018 ): 115-130 <https://dergipark.org.tr/en/pub/ieja/issue/33727/373650>
Chicago Ansari-toroghy, H. , Farshadifar, F. , Mahboobi-abkenar, F. "On the ideal-based zero-divisor graphs". International Electronic Journal of Algebra 23 (2018 ): 115-130
RIS TY - JOUR T1 - On the ideal-based zero-divisor graphs AU - HabibollahAnsari-toroghy, FaranakFarshadifar, FaridehMahboobi-abkenar Y1 - 2018 PY - 2018 N1 - doi: 10.24330/ieja.373650 DO - 10.24330/ieja.373650 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 115 EP - 130 VL - 23 IS - 23 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.373650 UR - https://doi.org/10.24330/ieja.373650 Y2 - 2022 ER -
EndNote %0 International Electronic Journal of Algebra On the ideal-based zero-divisor graphs %A Habibollah Ansari-toroghy , Faranak Farshadifar , Farideh Mahboobi-abkenar %T On the ideal-based zero-divisor graphs %D 2018 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 23 %N 23 %R doi: 10.24330/ieja.373650 %U 10.24330/ieja.373650
ISNAD Ansari-toroghy, Habibollah , Farshadifar, Faranak , Mahboobi-abkenar, Farideh . "On the ideal-based zero-divisor graphs". International Electronic Journal of Algebra 23 / 23 (January 2018): 115-130 . https://doi.org/10.24330/ieja.373650
AMA Ansari-toroghy H. , Farshadifar F. , Mahboobi-abkenar F. On the ideal-based zero-divisor graphs. IEJA. 2018; 23(23): 115-130.
Vancouver Ansari-toroghy H. , Farshadifar F. , Mahboobi-abkenar F. On the ideal-based zero-divisor graphs. International Electronic Journal of Algebra. 2018; 23(23): 115-130.
IEEE H. Ansari-toroghy , F. Farshadifar and F. Mahboobi-abkenar , "On the ideal-based zero-divisor graphs", International Electronic Journal of Algebra, vol. 23, no. 23, pp. 115-130, Jan. 2018, doi:10.24330/ieja.373650