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Graphical properties of the bipartite graph of Spec(Z[x])\{0}

Year 2015, Volume: 2 Issue: 1, 65 - 73, 22.01.2015
https://doi.org/10.13069/jacodesmath.66836

Abstract

Consider $Spec(Z[x])$, the set of prime ideals of $Z[x]$ as a partially ordered set under inclusion. By removing the zero ideal, we denote $G_{Z}=Spec(Z[x])\{0}$ and view it as an infinite bipartite graph with the prime ideals as the vertices and the inclusion relations as the edges. In this paper, we investigate fundamental graph theoretic properties of $G_{Z}$. In particular, we describe the diameter, circumference, girth, radius, eccentricity, vertex and edge connectivity, and cliques of $G_{Z}$. The complement of $G_{Z}$ is investigated as well.

References

  • D. F. Anderson and P. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217, 434-447, 1999.
  • E. Celikabs and C. Eubanks-Turner, The Projective Line over the Integers, Progress in Commutative Algebra II: Ring Theory, Homology, and Decompositions, 221-240, De Gruyter, 2012.
  • C. Eubanks-Turner, M. Luckas, S. Saydam, Prime ideals in Birational extensions of two-dimensional power series rings, Communications in Algebra, 41(2), 703-735, 2013.
  • W. Heinzer, C. Rotthaus, S. Wiegand, Mixed polynomial/power series rings and relations among their spectra, Multiplicative ideal theory in commutative algebra, Springer, New York, 227-242, 2006.
  • W. Heinzer, S. Wiegand, Prime ideals in two-dimensional polynomial rings, Proc. Amer. Math. Soc., 577-586, 1989.
  • W. Heinzer, S. Wiegand, Prime ideals in polynomial rings over one-dimensional domains, Trans. Amer. Math. Soc., 347(2), 639-650, 1995.
  • A. Li, S. Wiegand, The Polynomial Behavior of Prime Ideals in Polynomial Rings and the Projective Line over Z, Factorization in Integral Domains, Lecture Notes in Pure and Applied Mathematics, 189(3), 383-400, 1997.
  • A. Li, S. Wiegand, Prime ideals in two-dimensional domains over the integers, J. Pure Appl. Algebra, 130(3), 313-324, 1998.
  • D. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ, 2001.
  • R. Wiegand, Homeomorphisms of affine surfaces over a finite field, J. London Math. Soc., (2), 18(1), 28-32, 1978.
  • R. Wiegand, The prime spectrum of a two-dimensional affine domain, J. Pure Appl. Algebra, 40(2), 209-214, 1986.

Graphical properties of the bipartite graph of Spec(Z[x])\{0}

Year 2015, Volume: 2 Issue: 1, 65 - 73, 22.01.2015
https://doi.org/10.13069/jacodesmath.66836

Abstract

Consider $Spec(Z[x])$, the set of prime ideals of $Z[x]$ as a partially ordered set under inclusion. By removing the zero ideal, we denote $G_{Z}=Spec(Z[x])\{0}$ and view it as an infinite bipartite graph with the prime ideals as the vertices and the inclusion relations as the edges. In this paper, we investigate fundamental graph theoretic properties of $G_{Z}$. In particular, we describe the diameter, circumference, girth, radius, eccentricity, vertex and edge connectivity, and cliques of $G_{Z}$. The complement of $G_{Z}$ is investigated as well.

References

  • D. F. Anderson and P. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217, 434-447, 1999.
  • E. Celikabs and C. Eubanks-Turner, The Projective Line over the Integers, Progress in Commutative Algebra II: Ring Theory, Homology, and Decompositions, 221-240, De Gruyter, 2012.
  • C. Eubanks-Turner, M. Luckas, S. Saydam, Prime ideals in Birational extensions of two-dimensional power series rings, Communications in Algebra, 41(2), 703-735, 2013.
  • W. Heinzer, C. Rotthaus, S. Wiegand, Mixed polynomial/power series rings and relations among their spectra, Multiplicative ideal theory in commutative algebra, Springer, New York, 227-242, 2006.
  • W. Heinzer, S. Wiegand, Prime ideals in two-dimensional polynomial rings, Proc. Amer. Math. Soc., 577-586, 1989.
  • W. Heinzer, S. Wiegand, Prime ideals in polynomial rings over one-dimensional domains, Trans. Amer. Math. Soc., 347(2), 639-650, 1995.
  • A. Li, S. Wiegand, The Polynomial Behavior of Prime Ideals in Polynomial Rings and the Projective Line over Z, Factorization in Integral Domains, Lecture Notes in Pure and Applied Mathematics, 189(3), 383-400, 1997.
  • A. Li, S. Wiegand, Prime ideals in two-dimensional domains over the integers, J. Pure Appl. Algebra, 130(3), 313-324, 1998.
  • D. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ, 2001.
  • R. Wiegand, Homeomorphisms of affine surfaces over a finite field, J. London Math. Soc., (2), 18(1), 28-32, 1978.
  • R. Wiegand, The prime spectrum of a two-dimensional affine domain, J. Pure Appl. Algebra, 40(2), 209-214, 1986.
There are 11 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Christina Eubanks-turner This is me

Aihua Li This is me

Publication Date January 22, 2015
Published in Issue Year 2015 Volume: 2 Issue: 1

Cite

APA Eubanks-turner, C., & Li, A. (2015). Graphical properties of the bipartite graph of Spec(Z[x])\{0}. Journal of Algebra Combinatorics Discrete Structures and Applications, 2(1), 65-73. https://doi.org/10.13069/jacodesmath.66836
AMA Eubanks-turner C, Li A. Graphical properties of the bipartite graph of Spec(Z[x])\{0}. Journal of Algebra Combinatorics Discrete Structures and Applications. March 2015;2(1):65-73. doi:10.13069/jacodesmath.66836
Chicago Eubanks-turner, Christina, and Aihua Li. “Graphical Properties of the Bipartite Graph of Spec(Z[x])\{0}”. Journal of Algebra Combinatorics Discrete Structures and Applications 2, no. 1 (March 2015): 65-73. https://doi.org/10.13069/jacodesmath.66836.
EndNote Eubanks-turner C, Li A (March 1, 2015) Graphical properties of the bipartite graph of Spec(Z[x])\{0}. Journal of Algebra Combinatorics Discrete Structures and Applications 2 1 65–73.
IEEE C. Eubanks-turner and A. Li, “Graphical properties of the bipartite graph of Spec(Z[x])\{0}”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 2, no. 1, pp. 65–73, 2015, doi: 10.13069/jacodesmath.66836.
ISNAD Eubanks-turner, Christina - Li, Aihua. “Graphical Properties of the Bipartite Graph of Spec(Z[x])\{0}”. Journal of Algebra Combinatorics Discrete Structures and Applications 2/1 (March 2015), 65-73. https://doi.org/10.13069/jacodesmath.66836.
JAMA Eubanks-turner C, Li A. Graphical properties of the bipartite graph of Spec(Z[x])\{0}. Journal of Algebra Combinatorics Discrete Structures and Applications. 2015;2:65–73.
MLA Eubanks-turner, Christina and Aihua Li. “Graphical Properties of the Bipartite Graph of Spec(Z[x])\{0}”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 2, no. 1, 2015, pp. 65-73, doi:10.13069/jacodesmath.66836.
Vancouver Eubanks-turner C, Li A. Graphical properties of the bipartite graph of Spec(Z[x])\{0}. Journal of Algebra Combinatorics Discrete Structures and Applications. 2015;2(1):65-73.