JACODESMATH / ISSN 2148-838X

J. Algebra Comb. Discrete Appl.
2(1) • 65-73

Received: 3 August 2014; Accepted: 30 December 2014
DOI 10.13069/jacodesmath.66836

Journal of Algebra Combinatorics Discrete Structures and Applications

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

Research Article

Christina Eubanks-Turner1∗, Aihua Li2∗∗

1. Department of Mathematics, Loyola Marymount University, Los Angeles, California

2. Department of Mathematical Sciences, Montclair State University, Montclair, New Jersey

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 GZ = 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 GZ. In particular, we describe the diameter, circumference,
girth, radius, eccentricity, vertex and edge connectivity, and cliques of GZ. The complement of GZ is
investigated as well.

2010 MSC: 05C25, 05C63, 06A11

Keywords: Bipartite graph, Prime spectrum, Poset, Ring theory

1. Introduction

Throughout, R is a commutative Noetherian ring with identity 1. The prime spectrum of R, Spec(R),
is the partially ordered set (poset) of prime ideals of R ordered by inclusion. Considerable research has
been done in order to determine which partially ordered sets (posets) arise as spectra of two-dimensional
integral domains, such as, spectra of polynomial and power series rings [5–8, 10, 11].

In the past, diagrams were used to represent the prime spectra of certain rings. We refer to these
diagrams as spec graphs. For example, in 1989, William Heinzer and Sylvia Wiegand used spec graphs
to help illustrate the relationships of the elements of Spec(R[y]), where R is a semi-local countable one-
dimensional Noetherian domain and y is an indetereminate. Similarly, spec graphs have been used to
describe certain polynomial rings and power series rings (cf. [2–4, 7, 8]). The primary aim of those papers
is to find properties that the spectra satisfy as partially ordered sets, in particular those properties that
determine the posets.

∗ E-mail: ceturner@lmu.edu, supported by the Association for Women in Mathematics.
∗∗ E-mail: lia@mail.montclair.edu

65



Graphical properties of GZ

Recently, researchers have begun to give more attention to graphs derived from the ring structure.
They have also studied connections between the graph properties and the original rings. For example, in
1999 Anderson and Livingston introduced zero-divisor graphs [1]. In this direction we expand the idea
of applying graph theory to spec graphs.

For a ring R, we consider Spec(R) as an undirected graph whose vertices are the prime ideals of R
and two distinct vertices p and q are adjacent if and only if either p ⊂ q or q ⊂ p. Here the inclusion
“⊂” is strict, since Spec(R) is a poset. This implies there are no loops in the graph. We investigate
spec graphs of commutative rings in a graph theoretical way, which may lead to information about the
ring itself. In this paper we focus on the bipartite graph derived from Spec(Z[x]) \ {0}, which we call
GZ. We give background on the poset Spec(Z[x]) in Section 2. In Section 3, we give results describing
fundamental graph theoretic properties of GZ. Particularly, we show that the radius of GZ is 3, the
diameter of GZ is 4, GZ has infinite circumference and infinite vertex and edge connectivity. In Section
4 we look at certain interesting subgraphs of GZ and describe properties related to those subgraphs, in
particular, completeness. We also show that the complement, GZ, retains certain properties similar to
those of GZ, although GZ is not bipartite.

2. Overview of Spec(Z[x])

Definition 2.1. Let U be a partially ordered set with a unique minimal element u0. The height of an
element u in U, ht(u), is the length of the longest chain to u from u0. We denote the dimension of the
set U by dim(U) = max{ht(u)

∣∣ u ∈ U}. The set of height-i elements of U is denoted Hi(U) = {u ∈
U
∣∣ ht(u) = i}.

Notation 2.2. For U a partially ordered set and u ∈ U, we let u↑ = {v ∈ U|u < v} and u↓ = {v ∈
U|u > v}.

The spectrum of the polynomial ring Z[x] has been well described by Roger Wiegand in 1986 [11].
In [11] Roger Wiegand characterized Spec(Z[x]) as a partially ordered set satisfying five specific axioms,
called CZP Axioms.

Definition 2.3. [The CZP Axioms] For any poset U the following axioms are called the CZP Axioms:

(P1) U is countable with a unique minimal element.

(P2) U has dimension two.

(P3) Every height-one element of U is below countably infinitely many height-two elements.

(P4) For each pair of distinct elements of height one, there are only finitely many height-two ele-
ments above both of them.

(P5) Given finite subsets S and T, with ∅ 6= S ⊂ H1(U) and T ⊂ H2(U), there is a height-one
element w in U such that w < m, ∀m ∈ T, and whenever m′ ∈ U is greater than both w and s for
some s in S, then m′ ∈ T.

As in [8], we call the element w from (P5) the radical element of the pair (S, T) and it can be
generalized to higher dimension posets.

Definition 2.4. Let U be a partially ordered set of dimension n. Let ∅ 6= S ⊆ Hi(U) and T ⊆ Hi+1(U)
with 0 < i < n. An element w ∈ U is called a radical element if w < m and ∀m ∈ T, then w↑ ∩ s↑ ⊆ T,
for all s ∈ S.

Thus the fifth Axiom (P5) can be restated:

(P5′) Given a pair of finite subsets S, T of U such that ∅ 6= S ⊂ H1(U) and T ⊂ H2(U), there exists a
height-one radical element for the pair (S, T). We call such a pair (S, T) in (P5′) a (1, 2)-pair.

66



C. Eubanks-Turner, A. Li

Figure 1. Illustration of axiom (P5)

Theorem 2.5. If U is a poset satisfying (P1)-(P5) of Definition 2.3, then U ∼= Spec(Z[x]) [11].

The following Properties from [8] are useful for our investigation.

Theorem 2.6. Let U be a poset that satisfies the CZP axioms in Definition 2.3. Then

1. If w is a radical element for a (1, 2)-pair (S, T), then w /∈ S.

2. If S ⊆ S′ ⊆ H1(U) and T ⊆ T ′ ⊆ H2(U), where t ≯ s,∀t ∈ T ′ − T,∀s ∈ S 6= ∅, then a radical
element w for (S′, T ′) is also a radical element for (S, T).

3. In Spec(Z[x]), any (1, 2)-pair (S, T) has infinitely many radical elements.

Next we give a proposition useful for our investigation, which is related to Theorem 2.6.

Proposition 2.7. Let U be a poset that satisfies the CZP axioms.

1. ∀u ∈ H1(U), there exists infinitely many u′ ∈ H1(U) such that u↑ ∩u′↑ = ∅.

2. Every height-two element of U has infinitely many height-one elements below it.

3. Every height-two element of U has infinitely many height-one elements not below it.

Proof. 1. follows from Theorem 2.6: any radical element u′ for the (1, 2)-pair ({u},∅) will work. For
2. and 3., we use the fact that U ∼= Spec(Z[x]) as posets. Consider v = 〈f, p〉 ∈ Spec(Z[x]), where p is a
prime number and f is an irreducible polynomial in Z[x]. Then by Theorem 2.6 (3), there are infinitely
many radical elements for the (1, 2)-pair (〈p〉, v) and so we have Statement 2. For all prime integers q 6= p,
〈q〉 ∈ H1(U) and 〈q〉 ≮ v. Since there are infinitely many such q, 3 holds.

Theorem 2.8. The following Noetherian integral domains have spectra isomorphic to Spec(Z[x]):

1. The polynomial ring D[x], where D is an order in an algebraic number field [10];

2. Two-dimensional integral domains which are finitely generated algebras over an algebraic extension
of a finite field [11];

3. Finitely generated birational extensions of the polynomial ring Z[x] [8].

After characterizing Spec(Z[x]), Roger Wiegand gave the following conjecture: for every two-
dimensional finitely generated Z-algebra D, Spec(D) ∼= Spec(Z[x]), c.f. [10]. This conjecture remains
open.

67



Graphical properties of GZ

3. Properties of the graph GZ

In this section, we treat GZ := Spec(Z[x]) \ {0} as an undirected bipartite graph and investigate
basic graph theory properties related to GZ. First, we define the graph as follows:

Definition 3.1. Let U = Spec(Z[x]) be the poset of the prime ideals of Z[x] ordered by inclusion. The
graph GZ = (V (GZ), E(GZ)) derived from U, with partites H1(U), H2(U), is given by

V (GZ) = U \{0} = H1(U)∪H2(U) and

E(GZ) = {st | s ∈ H1(U), t ∈ H2(U), and s < t in U}.

All edges of GZ have the form: st, where s ∈ H1(U), t ∈ H2(U).

Remark 3.2. Note that since U is a partially ordered set this ensures the graph GZ is bipartite. That
is, every edge of GZ has one vertex in H1(U) and one vertex in H2(U).

We now denote Hi(U) as Hi(GZ) for i = 1, 2. Many basic graph theory properties of GZ can be
obtained from the ring theory properties of Spec(Z[x]), for example, Proposition 2.7 (2) and (3) shown
above. We investigate the connectivity, existence of cycles of different sizes, cut-edges, cut-vertices, values
of girth, diameter, circumference and so on. Existence of radical elements plays an important role in the
process. Here, we define the latter mentioned terms.

Definition 3.3. Let G be a graph.

1. The girth of G is the length of the shortest cycle contained in G.

2. The circumference of a graph G is the longest cycle contained in G.

3. A cut-edge in a graph G is an edge that increases the disconnectivity when removing it.

4. A cut-vertex in a graph G is a vertex that increases the disconnectivity when removing it.

Theorem 3.4. GZ has no cut-edges. Furthermore, every edge of GZ is contained in a cycle of length 4.
Furthermore, girth(GZ) = 4 and circumference(GZ) = ∞.

Proof. It is known that an edge is not a cut-edge if and only if it is included in a cycle. Assume
e = uv ∈ E(GZ), where u ∈ H1(GZ), v ∈ H2(GZ). By Theorem 2.5 (P3), there are infinitely many
elements in H2(GZ) adjacent to u. Choose v′ ∈ H2(GZ), where v 6= v′, such that uv′ ∈ E(GZ). By
Theorem 2.7 (3), there are infinitely many radical elements w ∈ H1(GZ) of the (1, 2)-pair ({u},{v, v′})
which are adjacent to each of v and v′, that is, wv, wv′ ∈ E(GZ). Thus we obtain infinitely many cycles
of length four of the form uvwv′u including the edge e = uv. So e is not a cut-edge. Furthermore, a
cycle of length 4 must be the shortest cycle since GZ is bipartite and there is no cycle of length 3. Thus,
girth(GZ) = 4.
To see that the circumference(GZ) = ∞, consider a vertex u1 ∈ H2(GZ). By Proposition 2.7 (2)
|u↓1| = ∞ and so there exist u2 ∈ H1(GZ) such that u1u2 ∈ E(GZ). We can choose u3 ∈ u

↑
2 \ {u1},

since |u↑2| = ∞ by Theorem 2.5 (P3). Note that in this setting u2, u4, . . . , u2i, . . . are height-one and
u1, u3, . . . , u2i+1, . . . are height-two. Continuing in this manner for any r ∈ N, choose a radical element
w := u2r for ({u2, . . . , u2r−2}, {u1, . . . , u2r−1}). Note u1, . . . , u2r−1 are all distinct. Thus we have a cycle
u1u2u3 · · ·u2r−1u2ru1 of length n = 2r. Since this holds for any n, where n is even, circumference(GZ) =
∞.

The next theorem states that GZ has no finite vertex cut-set.

Theorem 3.5. Consider the graph GZ. Then

68



C. Eubanks-Turner, A. Li

1. Every pair of vertices are connected by infinitely many paths of length ≤ 4.

2. GZ is connected.

3. There is no finite vertex cut-set of GZ, that is, the vertex-connectivity of GZ is infinite.

4. The edge-connectivity of GZ is infinite.

Proof. For 1, if uv ∈ E(GZ), u, v are connected by an edge, a path of length one. Moreover, by the
proof of Theorem 3.4, uv is contained in infinitely many cycles of length 4. Thus u and v are connected
by infinitely many paths of length 3. Next we assume u, v are not adjacent. We consider the following
cases.

Case 1: u, v ∈ H1(GZ).
By Theorem 2.5 (P3) we can choose t, t′ ∈ H2(GZ) such that ut, vt′ ∈ E(GZ) and t 6= t′. Thus we take
a radical element w for the pair ({u, v},{t, t′}). Obviously w /∈ {u, v}. So we have a path of length 4
connecting u and v: utwt′v. Note, there are infinitely many choices for such t and t′.

Case 2: u, v ∈ H2(GZ).
By Proposition 2.7 (2), there exist s ∈ H1(GZ) with su ∈ E(GZ). Similarly, there are infinitely many
radical elements w for the pair ({s},{u, v}) and we have the path uwv, of length 2.

Case 3: u ∈ H1(GZ) and v ∈ H2(GZ).
By Theorem 2.5 (P3), ∃t ∈ u↑\{v}. There are infinitely many radical elements w for the pair ({u},{t, v}),
which builds infinitely many paths of the form utwv of length 3.

Thus, 1 holds. The above also implies that every two vertices of GZ is connected. Thus GZ is
connected and statement 2 is true.

To see 3, let U′ be any finite subset of V (GZ) and consider u, v ∈ V (GZ) \ U′. By using the above
proof for 1, we can obtain infinitely many paths from u to v, not using any vertex from U′. Thus u and
v are connected in GZ \U′ and so U′ is not a vertex-cut set of GZ.

Item 3 implies 4 by the well-known fact that for any graph G, vertex connectivity of G ≤ edge
connectivity of G, see [9].

Next we give some basic concepts of graph theory and explore these parameters for GZ.

Definition 3.6. Let G be a graph.

1. The distance between two vertices u, v of G is the number of edges in a shortest path connecting
them. We denote the distance of u and v as d(u, v).

2. The eccentricity of a vertex v, ecc(v), is the greatest distance from v to any other vertex.

3. The radius of a graph is the minimum eccentricity over all vertices in the graph. We denote the
radius of G as rad(G). A central vertex in a graph is a vertex that achieves the radius.

4. The diameter of a graph is the maximum eccentricity over all the vertices in the graph. That is, it
is the greatest distance over all pairs of vertices. We denote the diameter of a graph G as diam(G).
A peripheral vertex in a graph is a vertex that achieves the diameter.

Theorem 3.7. GZ satisfies the following properties:

1. For u, v ∈ V (GZ), d(u, v) ≤ 4.

2. ecc(v) =

{
3, if v ∈ H2(GZ)
4, if v ∈ H1(GZ)

3. rad(GZ) = 3.

69



Graphical properties of GZ

4. diam(GZ) = 4.

5. H1(GZ) = {all peripheral vertices of GZ} and H2(GZ) = {all central vertices of GZ}.

6. GZ has no finite maximal path.

Proof. 1. and 2. follow from the proof of Theorem 3.5. It is straightforward that 2. implies 3,4
and 5. To see 6., let Pn = u1 . . . un+1 be a path of length n ≥ 1. If un+1 ∈ H1(GZ), we choose
un+2 ∈ u

↑
n+1\{u1, . . . , un} to extend Pn into a path u1 . . . un+1un+2 of length n+1. The existence of un+2

is based on the fact that |u↑n+1| = ∞ by Theorem 2.5 (P3). If un+1 ∈ H2(GZ), ∃un+2 ∈ u
↓
n+1\{u1, . . . , un}

by Proposition 2.7 (2). A longer path is produced again: u1 . . . un+1un+2.

Another topic that generates a lot of interest in graph theory is the concept of graph matchings.
Below we define what a matching is and show that GZ has a perfect matching.

Definition 3.8. A matching in a graph G is a set of pair-wise vertex disjoint edges. A perfect matching
is a matching which covers all vertices of the graph. That is, every vertex of the graph is incident to
exactly one edge of the matching.

Theorem 3.9. GZ has a perfect matching.

Proof. We build a matching step by step that can cover all the vertices of GZ. Start with any edge
u1v1 of GZ, which is a matching of size 1, where u1 ∈ H1(GZ) and v1 ∈ H2(GZ). Call it M1 = {u1v1}.
We then list all the rest of the vertices of H1(GZ) in a countable way: u2, u3, . . .. Similarly, we build
the countable list for the rest of the vertices of H2(GZ): v2, v3, . . . .. Next we pick u2 ∈ H1(GZ) and
choose v′2 from u

↑
2 \{v1}. This is doable because

∣∣∣u↑2∣∣∣ = ∞. Now we have a matching M2 = {u1v1, u2v′2}
of size 2. In the next step, we will pick v2 and find u′3 ∈ v

↓
2 \
{
u1, u2

}
to extend the mating M2 into

M3 =
{
u1v1, u2v

′
2, v2u

′
3, . . .

}
. Continue picking vertices alternatively from the countable lists of the two

partites to enlarge the matching that eventually can cover all the vertices of GZ.

4. Completeness in GZ and the complement of GZ

A complete bipartite graph is a bipartite graph where every vertex of the first partite is adjacent to
every vertex of the second set. We adopt the commonly used notation Km,n for the complete bipartite
graph with vertex partitions of size m and n. Note that for any n ∈ N, K1,n ⊆ GZ, by (P3) Theorem 2.5.
Also we consider the complement graph of GZ and give some interesting properties of it.

4.1. Completeness

Here, we give a theorem showing that there are infinitely many finite complete subgraphs of GZ.

Theorem 4.1. For every m, n ∈ N, there are countably infinitely many subgraphs of GZ which are
isomorphic to Km,n.

Proof. Without loss of generality, assume m ≤ n. We first consider the corresponding situation
in Spec(Z[x]). Let p1, p2, . . . , pn be distinct prime numbers. Then xi + p1p2 . . . pn is irreducible in
Z[x],∀i, 1 ≤ i ≤ n. So 〈xi + p1p2 . . . pn〉 ∈ H1(Spec(Z[x])), for each i. Let S = {〈xi + p1p2 . . . pn〉}ni=1 and
T = {〈x, pj〉}mj=1 ⊆ H2(Spec(Z[x])). Then 〈xi + p1p2 . . . pn〉 ⊂ 〈x, pj〉, for all i, j, 1 ≤ i ≤ n, 1 ≤ j ≤ m.
Thus the corresponding subgraph of GZ induced by S ∪ T is complete and isomorphic to Km,n. Since
there are infinitely many choices for such p1, p2, . . . , pn, we have infinitely many of these subgraphs.

70



C. Eubanks-Turner, A. Li

An immediate Corollary of Theorem 4.1 is that GZ is not planar. First, recall that a graph G is
planar if G can be embedded into a plane. That is, G can be drawn on a plane in such a way that
edges only intersect at their common vertices. Also recall that a graph is planar if and only if it does not
contain K3,3 or K5 [9].

Corollary 4.2. GZ is not planar and it has infinitely many non-planar subgraphs.

Proof. By Theorem 4.1, there are infinitely subgraphs of GZ isomorphic to K3,3. Thus we have the
claim by Kuratowski’s Theorem, see [9].

Next, we give a proposition that shows we can extend any finite complete bipartite subgraph of GZ
into a larger complete bipartite subgraph of GZ. Recall that for a graph G, a clique of a graph G is a
complete subgraph of G.

Proposition 4.3. Let Kn,m be a complete bipartite subgraph of GZ with partites S ⊆ H1(GZ) and
T ⊆ H2(GZ), n, m ≥ 1.

1. Let r1 be a positive integer. Then Kn,m can be extended to a complete subgraph of GZ that is
isomorphic to Kn+r1,m.

2. Let S↑ =
⋂
s∈S

s↑. Assume |S↑| = r0 > m. Then for every integer r2, 0 < r2 ≤ r0 −m, Kn,m can be

extended to a subgraph that is isomorphic to Kn,m+r2.

3. GZ has no maximum clique. However, for a fixed finite clique in GZ, all the extended cliques have
a maximum partite in H2(GZ).

Proof. For 1., choose a radical element w1 for the (1, 2)-pair (S, T). Let T↓ =
⋂
t∈T

t↓. Since w1 ∈ T↓,

the subgraph of GZ induced by S∪{w1}∪T is a complete bipartite subgraph. Continuing in this process,
we can choose distinct w1, w2, . . . , wr1 not in S∪T such that S∪{w1, w2, . . . , wr1}∪T generates a clique
of GZ isomorphic to Kn+r1,m. To see 2., we can choose T0 ⊆ S↑\T with |T0| = r2. Then S ∪T0 ∪T will
induce a clique of GZ which is isomorphic to Kn,m+r2.

For 3, without loss of generality, we assume S has at least two vertices. From the proof of part
2, any added vertex from H2(GZ) in an enlarged clique must be in S↑ \ T , which is finite by the CZP
Axiom (P4) since |S| ≥ 2. Thus only finitely many vertices from H2(GZ) can be added to any extended
clique.

4.2. The complement of GZ

Recall that the complement G of a graph G is a graph with the same vertex set as G such that two
vertices u, v are adjacent in G if and only if u, v are not adjacent in G. Since GZ is bipartite, the two
partites of GZ induce two complete subgraphs of GZ. Proposition 4.5 asserts that GZ is connected. In
GZ, we denote NGZ(u) = {v ∈ V (GZ)|uv ∈ E(GZ)} = {v ∈ V (GZ)|uv 6∈ E(GZ)}.

Proposition 4.4. For every u ∈ V (GZ), |NGZ(u)| = ∞.

Proof. Case 1. u ∈ H1(GZ). Note that NGZ(u) = H2(GZ) \ u
↑. Assume NGZ(u) is finite. Then

T = H2(GZ) \ u↑ is finite. Let S = {u} and consider the (1, 2)-pair (S, T) in GZ. Since both S, T are
finite and S 6= ∅, there exists a radical element w for the pair (S, T). Also since w ∈ H1(GZ), |w↑| = ∞,
and H2(GZ) = u↑ ∪T , ∃m ∈ w↑ ∩u↑ which implies m /∈ T , contradiction.

Case 2. u ∈ H2(GZ). By Proposition 1(3), there are infinitely many vertices in H1(GZ) not adjacent
to u in GZ thus adjacent to u in GZ.

Thus in both cases, |NGZ(u)| = ∞.

71



Graphical properties of GZ

Proposition 4.5. The complement GZ is connected and both the vertex and edge connectivity of GZ are
infinite.

Proof. It is obvious that GZ is made of two complete subgraphs induced from the vertex sets H1(GZ)
and H2(GZ) and there are infinitely many edges connecting vertices between the two cliques. Thus GZ
is connected.

To see that GZ has infinite vertex-connectivity, let U′ be a finite vertex cut set and consider any two
vertices u, v ∈ V (GZ) \ U′. We show that u and v are connected in GZ. Since each of H1(GZ), H2(GZ)
induces the empty graph in GZ , we have that each induces a complete subgraph of GZ. It is sufficient
to assume u ∈ H1(GZ) and v ∈ H2(GZ). By Proposition 4.4 since

∣∣∣NGZ(v)∣∣∣ = ∞, and ∣∣∣U′∣∣∣ < ∞, we can
choose w ∈ NGZ(v)\U

′ to make a cycle uwvu in GZ. Since there are infinitely many such w, v and u are
connected in V (GZ)\U′, we have U′ is not a cut-set of GZ. Thus, GZ has infinite vertex-connectivity. The
latter implies infinite edge-connectivity, by the well-known fact that for any graph G, vertex connectivity
of G ≤ edge connectivity of G, see [9].

5. What does the graph GZ reveal about the ring Z[x]?

Much research has been done in describing the structures of rings whose spectra are isomorphic to
Spec(Z[x]) as posets. The work in this paper gives a set of graph theoretic properties of the derived graph
GZ by Spec(Z[x]) \ {0} as an infinite bipartite graph. Is it shown that GZ is a graph with many good
properties. Although it is infinite, it has fairly small graph measurements such as girth, diameter, radius,
etc. It contains many different types of finite bipartite subgraphs such as perfect matching, cliques of
any size, etc. It is natural to ask: “What does the graph GZ reveal about the ground rings represented
by Z[x]?" In this regard, we give some descriptions from the ring theoretic point of view about any ring
whose spectrum satisfies the CZP Axioms, based on the graph theory results obtained in earlier sections
of this paper.

Proposition 5.1. Let D be a ring such that Spec(D) satisfies the CZP axioms given in Definition 2, as
a poset. Then the following are true:

1. Assume p1, p2 are two co-maximal height-1 prime ideals (so p1 + p2 = D). Then there exist a
height-1 prime p and two maximal ideals m1, m2 such that pi + p ⊆ mi for i = 1, 2.

2. For any pair (p, m) of prime ideals, where p is of height one, m is maximal, and p 6⊆ m, there exisit
a height-1 prime p′ and a maximal ideal m′ such that p ⊆ m′ and p′ ⊆ m∩m′.

3. We can order all the height-1 ideals and all the maximal ideals as two corresponding countable lists:

H2(Spec(D)) : m1, m2, . . . , mn, . . .

H1(Spec(D)) : p1, p2, . . . , pn, . . .

such that for each i, pi ⊆ mi.

4. For any positive integers n and r, there are n height-1 prime ideals
p1, p2, . . . , pn and r maximal ideals m1, m2, . . . , mr such that pi ⊆ mj for all i, j where 1 ≤ i ≤ n
and 1 ≤ j ≤ r.

Proof. Statement 1 is immediately from Theorem 3.5 (1) (refer to Case 1 in the proof). Statement
2 is also by Theorem 3.5 (1) (refer to Case 3 in the proof). Statement 3 is from Theorem 3.9: GZ has
a perfect matching. The last statement follows directly from the existence of cliques of any size in GZ
(Theorem 4.1).

72



C. Eubanks-Turner, A. Li

6. Conclusion

This research shows that graphs derived from ring spectra may provide rich structural properties
and may contain various types of graphs of interest, thus worthy of study. The graphic properties may
tell something about the ring structure itself and vice versa. In the future, we will examine more types
of spectra and apply more advanced graph theory techniques, such as graph polynomials, in the study.

References

[1] D. F. Anderson and P. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217,
434-447, 1999.

[2] 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.

[3] 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.

[4] 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.

[5] W. Heinzer, S. Wiegand, Prime ideals in two-dimensional polynomial rings, Proc. Amer. Math. Soc.,
577-586, 1989.

[6] W. Heinzer, S. Wiegand, Prime ideals in polynomial rings over one-dimensional domains, Trans.
Amer. Math. Soc., 347(2), 639-650, 1995.

[7] 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.

[8] A. Li, S. Wiegand, Prime ideals in two-dimensional domains over the integers, J. Pure Appl. Algebra,
130(3), 313-324, 1998.

[9] D. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ, 2001.
[10] R. Wiegand, Homeomorphisms of affine surfaces over a finite field, J. London Math. Soc., (2), 18(1),

28-32, 1978.
[11] R. Wiegand, The prime spectrum of a two-dimensional affine domain, J. Pure Appl. Algebra, 40(2),

209-214, 1986.

73


	Introduction
	Overview of Spec(Z[x])
	Properties of the graph GZ
	Completeness in GZ and the complement of GZ
	 What does the graph GZ reveal about the ring Z[x]?
	Conclusion
	References