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Year 2020, Volume 28, Issue 28, 61 - 74, 14.07.2020
https://doi.org/10.24330/ieja.768135

Abstract

References

  • D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra, 159 (1993), 500-514.
  • D. F. Anderson and A. Badawi, On the zero-divisor graph of a ring, Comm. Algebra, 36 (2008), 3073-3092.
  • D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320 (2008), 2706-2719.
  • D. F. Anderson and A. Badawi, On the total graph of a commutative ring without the zero element, J. Algebra Appl., 11(4) (2012), 1250074 (18 pp).
  • D. F. Anderson and A. Badawi, The generalized total graph of a commutative ring, J. Algebra Appl., 12(5) (2013), 1250212 (18 pp).
  • D. F. Anderson and J. D. LaGrange, The semilattice of annihilator classes in a reduced commutative ring, Comm. Algebra, 43 (2015), 29-42.
  • D. F. Anderson and J. D. LaGrange, Some remarks on the compressed zero-divisor graph, J. Algebra, 447 (2016), 297-321.
  • D. F. Anderson and E. F. Lewis, A general theory of zero-divisor graphs over a commutative ring, Int. Electron. J. Algebra, 20 (2016), 111-135.
  • D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434-447.
  • D. F. Anderson and S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra, 210 (2007), 543-550.
  • D. F. Anderson and D. Weber, The zero-divisor graph of a commutative ring without identity, Int. Electron. J. Algebra, 23 (2018), 176-202.
  • S. Akbari, H. R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra, 270 (2003), 169-180.
  • S. Akbari, D. Kiani, F. Mohammadi and S. Moradi, The total graph and regular graph of a commutative ring, J. Pure Appl. Algebra, 213 (2009), 2224-2228.
  • A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra, 42 (2014), 108-121.
  • A. Badawi, On the dot product graph of a commutative ring, Comm. Algebra, 43 (2015), 43-50.
  • I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988), 208-226.
  • C. F. Kimball and J. D. LaGrange, The idempotent-divisor graphs of a commutative ring, Comm. Algebra, 46 (2018), 3899-3912.
  • J. D. LaGrange, The x-divisor pseudographs of a commutative groupoid, Int. Electron. J. Algebra, 22 (2017), 62-77.
  • W. J. LeVeque, Fundamentals of Number Theory, Addison-Wesley Publishing Company, Reading, Massachusetts, 1977.
  • S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra, 30 (2002), 3533-3558.
  • R. Nikandish, M. J. Nikmehr and M. Bakhtyiari, Coloring of the annihilator graph of a commutative ring, J. Algebra Appl., 15(7) (2016), 1650124 (13 pp).
  • Z. Pucanovic and Z. Petrovic, On the radius and the relation between the total graph of a commutative ring and its extensions, Publ. Inst. Math. (Beograd) (N.S.), 89 (2011), 1-9.
  • P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176 (1995), 124-127.
  • M. Sivagami and T. Tamizh Chelvam, On the trace graph of matrices, Acta Math. Hungar., 158 (2019), 235-250.
  • S. Spiroff and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra, 39 (2011), 2338-2348.
  • T. Tamizh Chelvam and T. Asir, On the genus of the total graph of a commutative ring, Comm. Algebra, 41 (2013), 142-153.

ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II

Year 2020, Volume 28, Issue 28, 61 - 74, 14.07.2020
https://doi.org/10.24330/ieja.768135

Abstract

In 2015, the second-named author introduced the dot product graph associated to a commutative ring $A$. Let $A$ be a commutative ring with nonzero identity, $1 \leq n < \infty$ be an integer, and $R = A \times A \times \cdots \times A$ ($n$ times). We recall that the total dot product graph of $R$ is the (undirected) graph $TD(R)$ with vertices $R^* = R\setminus \{(0, 0, \dots, 0)\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $x\cdot y = 0 \in A$ (where $x\cdot y$ denotes the normal dot product of $x$ and $y$). Let $Z(R)$ denote the set of all zero-divisors of $R$. Then the zero-divisor dot product graph of $R$ is the induced subgraph $ZD(R)$ of $TD(R)$ with vertices $Z(R)^* = Z(R) \setminus \{(0, 0, \dots, 0)\}$. Let $U(R)$ denote the set of all units of $R$. Then the unit dot product graph of $R$ is the induced subgraph $UD(R)$ of $TD(R)$ with vertices $U(R)$. In this paper, we study the structure of $TD(R)$, $UD(R)$, and $ZD(R)$ when $A = Z_n$ or $A = GF(p^n)$, the finite field with $p^n$ elements, where $n \geq 2$ and $p$ is a prime positive integer.

References

  • D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra, 159 (1993), 500-514.
  • D. F. Anderson and A. Badawi, On the zero-divisor graph of a ring, Comm. Algebra, 36 (2008), 3073-3092.
  • D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320 (2008), 2706-2719.
  • D. F. Anderson and A. Badawi, On the total graph of a commutative ring without the zero element, J. Algebra Appl., 11(4) (2012), 1250074 (18 pp).
  • D. F. Anderson and A. Badawi, The generalized total graph of a commutative ring, J. Algebra Appl., 12(5) (2013), 1250212 (18 pp).
  • D. F. Anderson and J. D. LaGrange, The semilattice of annihilator classes in a reduced commutative ring, Comm. Algebra, 43 (2015), 29-42.
  • D. F. Anderson and J. D. LaGrange, Some remarks on the compressed zero-divisor graph, J. Algebra, 447 (2016), 297-321.
  • D. F. Anderson and E. F. Lewis, A general theory of zero-divisor graphs over a commutative ring, Int. Electron. J. Algebra, 20 (2016), 111-135.
  • D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434-447.
  • D. F. Anderson and S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra, 210 (2007), 543-550.
  • D. F. Anderson and D. Weber, The zero-divisor graph of a commutative ring without identity, Int. Electron. J. Algebra, 23 (2018), 176-202.
  • S. Akbari, H. R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra, 270 (2003), 169-180.
  • S. Akbari, D. Kiani, F. Mohammadi and S. Moradi, The total graph and regular graph of a commutative ring, J. Pure Appl. Algebra, 213 (2009), 2224-2228.
  • A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra, 42 (2014), 108-121.
  • A. Badawi, On the dot product graph of a commutative ring, Comm. Algebra, 43 (2015), 43-50.
  • I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988), 208-226.
  • C. F. Kimball and J. D. LaGrange, The idempotent-divisor graphs of a commutative ring, Comm. Algebra, 46 (2018), 3899-3912.
  • J. D. LaGrange, The x-divisor pseudographs of a commutative groupoid, Int. Electron. J. Algebra, 22 (2017), 62-77.
  • W. J. LeVeque, Fundamentals of Number Theory, Addison-Wesley Publishing Company, Reading, Massachusetts, 1977.
  • S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra, 30 (2002), 3533-3558.
  • R. Nikandish, M. J. Nikmehr and M. Bakhtyiari, Coloring of the annihilator graph of a commutative ring, J. Algebra Appl., 15(7) (2016), 1650124 (13 pp).
  • Z. Pucanovic and Z. Petrovic, On the radius and the relation between the total graph of a commutative ring and its extensions, Publ. Inst. Math. (Beograd) (N.S.), 89 (2011), 1-9.
  • P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176 (1995), 124-127.
  • M. Sivagami and T. Tamizh Chelvam, On the trace graph of matrices, Acta Math. Hungar., 158 (2019), 235-250.
  • S. Spiroff and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra, 39 (2011), 2338-2348.
  • T. Tamizh Chelvam and T. Asir, On the genus of the total graph of a commutative ring, Comm. Algebra, 41 (2013), 142-153.

Details

Primary Language English
Subjects Mathematics
Journal Section Articles
Authors

Mohammad ABDULLA This is me
The American University of Sharjah
United Arab Emirates


Ayman BADAWI This is me (Primary Author)
The American University of Sharjah
United Arab Emirates

Publication Date July 14, 2020
Published in Issue Year 2020, Volume 28, Issue 28

Cite

Bibtex @research article { ieja768135, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2020}, volume = {28}, number = {28}, pages = {61 - 74}, doi = {10.24330/ieja.768135}, title = {ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II}, key = {cite}, author = {Abdulla, Mohammad and Badawı, Ayman} }
APA Abdulla, M. & Badawı, A. (2020). ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II . International Electronic Journal of Algebra , 28 (28) , 61-74 . DOI: 10.24330/ieja.768135
MLA Abdulla, M. , Badawı, A. "ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II" . International Electronic Journal of Algebra 28 (2020 ): 61-74 <https://dergipark.org.tr/en/pub/ieja/issue/55997/768135>
Chicago Abdulla, M. , Badawı, A. "ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II". International Electronic Journal of Algebra 28 (2020 ): 61-74
RIS TY - JOUR T1 - ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II AU - MohammadAbdulla, AymanBadawı Y1 - 2020 PY - 2020 N1 - doi: 10.24330/ieja.768135 DO - 10.24330/ieja.768135 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 61 EP - 74 VL - 28 IS - 28 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.768135 UR - https://doi.org/10.24330/ieja.768135 Y2 - 2022 ER -
EndNote %0 International Electronic Journal of Algebra ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II %A Mohammad Abdulla , Ayman Badawı %T ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II %D 2020 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 28 %N 28 %R doi: 10.24330/ieja.768135 %U 10.24330/ieja.768135
ISNAD Abdulla, Mohammad , Badawı, Ayman . "ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II". International Electronic Journal of Algebra 28 / 28 (July 2020): 61-74 . https://doi.org/10.24330/ieja.768135
AMA Abdulla M. , Badawı A. ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II. IEJA. 2020; 28(28): 61-74.
Vancouver Abdulla M. , Badawı A. ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II. International Electronic Journal of Algebra. 2020; 28(28): 61-74.
IEEE M. Abdulla and A. Badawı , "ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II", International Electronic Journal of Algebra, vol. 28, no. 28, pp. 61-74, Jul. 2020, doi:10.24330/ieja.768135