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.
Keywords
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.
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.
Keywords
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.
Details
| Primary Language | English |
|---|---|
| Journal Section | Articles |
| Authors | |
| Publication Date | January 22, 2015 |
| Published in Issue | Year 2015 |