THE NONORIENTABLE GENUS OF SOME JACOBSON GRAPHS AND CLASSIFICATION OF THE PROJECTIVE ONES

H. AMRAEI, H. R. MAIMANI, M. R. POURNAKI, AND A. ZAEEMBASHI

Abstract. Let R be a finite commutative ring with nonzero identity and denote its Jacobson radical by J(R). The Jacobson graph of R is the graph in which the vertex set is R \ J(R), and two distinct vertices x and y are adjacent if and only if 1 − xy is not a unit in R. In this paper, the nonorientable genus of some Jacobson graphs is either computed or estimated by a lower bound. As an application, the rings R with projective Jacobson graphs are classified.

1. Introduction

A surface is a real 2-manifold, that is, a connected Hausdorff topological space with a countable basis such that every point lies in an open neighborhood homeomorphic to an open neighborhood of ℝ^{2}. A nonorientable surface is a surface that contains a subspace which is homeomorphic to a Möbius strip. A connected sum of two surfaces is the surface obtained by cutting an open disc out of each surface and then gluing the two surfaces together along the boundaries of the holes. For the positive integer q, N_{q} denotes the surface formed by a connected sum of q projective planes. It is easy to see that N_{q} is a nonorientable surface. We may denote the sphere by N_{0}, even though it is orientable. The surface N_{1} is the projective plane and the surface N_{2} is the Klein bottle. The nonnegative integer q is called the nonorientable genus of N_{q}. The nonorientable genus of a finite graph G without loops or multiple edges, denoted by γ̄(G), is the smallest nonnegative integer q for which there exists an embedding of G into N_{q}. The graphs G with γ̄(G) = 0 are in fact planar graphs and the graphs G with γ̄(G) = 1 are called projective graphs. We refer the reader to [8] for a treatment of the subject.

In this paper, we deal with some graphs arising from rings. These graphs are called Jacobson graphs. We compute the nonorientable genus of some of them and give a lower bound for the some other. As an application, we classify, up to isomorphism, the finite commutative rings with nonzero identity whose Jacobson graphs are projective. It is noticeable that associating a graph to an algebraic structure is a research subject in the area of algebraic combinatorics and has attracted considerable attention. The research in this subject aims at exposing the relationship between algebra and graph theory and at advancing the application of one to the other. In fact, there are three major problems in this subject: (1) characterization of the resulting graphs, (2) characterization of the algebraic structures with isomorphic graphs, and (3) realization of the connections between the algebraic structures and the corresponding graphs. In literature, one can find a number of different types of graphs attached to rings or other algebraic structures. We refer the reader to [2] and [7] for a survey of recent results concerning graphs attached to rings.

2000 Mathematics Subject Classification. Primary: 05C75; Secondary: 13M05.

Key words and phrases. Finite commutative ring, Jacobson graph, Nonorientable genus.

The research of H. R. Maimani and M. R. Pournaki was in part supported by a grant from IPM (No. 94050113 and No. 94130115).

2. Preliminaries

In this section, we recall some definitions and notations concerning graphs for later use. In the sequel, by a graph we mean a finite graph without loops or multiple edges. Also, for the convenience of the reader, we state without proof a few known results in the form of propositions which will be used later. For unexplained terminology and notations concerning graph theory we refer the reader to [4].

The present paper deals with what is known as the Jacobson graph of a ring. Some of the properties of this graph have been studied in detail in [3]. First, let us state the definition of Jacobson graphs.

Definition 2.1. Let R be a finite commutative ring with nonzero identity, J(R) denote the Jacobson radical of R and let U(R) be the set of unit elements in R. The Jacobson graph of R, denoted by 𝔍_{R}, is defined to be the graph in which the vertex set is R\J(R), and two distinct vertices x and y are adjacent if and only if 1−xy ∉ U(R).

The graphs in Figure 1 are the Jacobson graphs of the rings indicated. Throughout the paper, in all the figures, we abbreviate the ordered pair (r, s) by rs.

[figure: 1; 3; 𝔍_{ℤ_{4}}; 11; 10; 01; 𝔍_{ℤ_{2}×ℤ_{2}}; 01; 10; 12; 11; 02; 𝔍_{ℤ_{2}×ℤ_{3}}]

Figure 1. The Jacobson graphs of some specific rings.

The following result of paper [3] characterizes the planar Jacobson graphs which will be used later frequently. Note that the ring ℤ_{2} × \frac{ℤ_{2}[x]}{(x^{2})} is missed in the paper [3]

and 𝔽_{4} denotes the ring \frac{ℤ_{2}[x]}{(x^{2}+x+1)}.

Proposition 2.2 ([3], Theorem 4.3). Let R be a finite commutative ring with nonzero identity. Then the Jacobson graph 𝔍_{R} is a planar graph if and only if either R is a field or it is isomorphic to one of ℤ_{4}, ℤ_{8}, ℤ_{9}, ℤ_{2} × ℤ_{2}, ℤ_{2} × ℤ_{3}, ℤ_{2} × ℤ_{4}, ℤ_{2} × 𝔽_{4}, ℤ_{2} × \frac{ℤ_{2}[x]}{(x^{2})}, ℤ_{3} × ℤ_{3}, ℤ_{2} × ℤ_{2} × ℤ_{2}, \frac{ℤ_{2}[x]}{(x^{2})}, \frac{ℤ_{2}[x]}{(x^{3})}, \frac{ℤ_{3}[x]}{(x^{2})}, \frac{ℤ_{4}[x]}{(2x,x^{2})}, \frac{ℤ_{4}[x]}{(2x,x^{2}−2)} or \frac{ℤ_{2}[x,y]}{(x,y)^{2}}.

We now state the following proposition which is needed for later use. Let us recall that a graph G is said to be complete if there is an edge between every pair of distinct vertices in G. We denote the complete graph with n vertices by K_{n}. A bipartite graph is the one whose vertex set can be partitioned into two disjoint parts in such a way that the two end vertices of every edge lie in different parts. Among the bipartite graphs, the complete bipartite graph is the one in which two vertices are adjacent if and only if they lie in different parts. The complete bipartite graph, with parts of size m and n, is denoted by K_{m,n}.

Proposition 2.3 ([3], Theorem 2.2). Let R be a finite commutative ring with nonzero identity. If R is local with maximal ideal 𝔪 and residue field 𝕂 = R/𝔪, then the connected components of 𝔍_{R} are isomorphic either to complete graph K_{|𝔪|} or to complete bipartite graph K_{|𝔪|,|𝔪|}. Moreover,

(1) if |𝕂| is odd, then 𝔍_{R} has two complete components and (|𝕂| − 3)/2 complete bipartite components, and

(2) if |𝕂| is even, then 𝔍_{R} has one complete component and (|𝕂| − 2)/2 complete bipartite components.

The following two propositions give us the nonorientable genus of the complete and complete bipartite graphs.

Proposition 2.4 ([10], Theorem 11–19). If n ≥ 3, then

γ̄(K_{n}) = \{ \begin{matrix}⌈\frac{(n − 3)(n − 4)}{6}⌉ & if n ≠ 7, \\ 3 & if n = 7. \end{matrix}

Proposition 2.5 ([10], Theorem 11–23). If m, n ≥ 2, then

γ̄(K_{m,n}) = ⌈\frac{(m − 2)(n − 2)}{2}⌉ .

The following two propositions give us a lower bound for the nonorientable genus of connected graphs.

Proposition 2.6 ([10], Theorem 11–7). If G is a connected graph with n ≥ 3 vertices and m edges, then γ̄(G) ≥ \frac{m}{3} − n + 2.

Proposition 2.7 ([10], Theorem 11–8). If G is a connected graph with n ≥ 3 vertices and m edges and has no triangles, then γ̄(G) ≥ \frac{m}{2} − n + 2.

Given a graph G, we denote its vertex set by V (G) and its edge set by E(G). If G_{1} and G_{2} are any two graphs, then their disjoint union, denoted by G_{1} ⊔ G_{2}, is defined to be the graph in which the vertex set is V (G_{1}) ⊔ V (G_{2}) and the edge set is E(G_{1}) ⊔ E(G_{2}). The following result, which follows from [5, Theorem 1.4], often enables us to reformulate some results which are otherwise true for connected graphs. Let us recall that the genus of a graph G, denoted by γ(G), is the smallest nonnegative integer g such that the graph G can be embedded on the surface obtained by attaching g handles to a sphere.

Proposition 2.8 ([5], Theorem 1.4). If a graph G is isomorphic to the disjoint union G_{1} ⊔G_{2} of two graphs G_{1} and G_{2}, then γ̄(G_{1} ⊔G_{2}) = γ̄(G_{1})+γ̄(G_{2})+δ, where δ = −1 if either γ̄(G_{1}) > 2γ(G_{1}) or γ̄(G_{2}) > 2γ(G_{2}), and δ = 0 otherwise.

3. The Main Theorem

Rings whose Jacobson graphs are either planar or toroidal have been already classified (see [3, Theorem 4.3] and [1, Theorem 1.1]). As the main result of this paper, we classify, up to isomorphism, the rings whose Jacobson graphs are projective. We recall that, in the sequel, 𝔽_{4} denotes the ring \frac{ℤ_{2}[x]}{(x^{2}+x+1)}.

Main Theorem. Let R be a finite commutative ring with nonzero identity. Then the Jacobson graph 𝔍_{R} is a projective graph if and only if R is isomorphic to one of ℤ_{2} × ℤ_{5}, ℤ_{3} × ℤ_{4}, ℤ_{3} × 𝔽_{4} or ℤ_{3} × \frac{ℤ_{2}[x]}{(x^{2})}.

We divide the proof of the Main Theorem into a series of lemmas. Let us start with the following one.

Lemma 3.1. Let R be a finite commutative ring with nonzero identity having a maximal ideal of size ≥ 7. Then γ̄(𝔍_{R}) ≥ 3.

Proof. By the assumption, R has at least a maximal ideal 𝔪 = {m_{1}, …, m_{ℓ}} with ℓ ≥ 7. Let i ≠ j with 1 ≤ i, j ≤ ℓ be given. Note that 1 + m_{i} and 1 + m_{j} are distinct elements of R \ J(R), and 1 − (1 + m_{i})(1 + m_{j}) ∉ U(R). Therefore, 1 + m_{i} and 1 + m_{j} are adjacent vertices in 𝔍_{R}. This implies that the vertices 1 + m_{1}, …, 1 + m_{ℓ} are mutually adjacent in the Jacobson graph 𝔍_{R} and so it has a subgraph isomorphic to K_{ℓ}, where ℓ ≥ 7. Now, Proposition 2.4 implies that γ̄(𝔍_{R}) ≥ 3. □

We also need to characterize all finite commutative rings with nonzero identity which are either a local ring, a product of two local rings or a product of three local rings, and all of their maximal ideals have size ≤ 6. In the following lemma, we give this characterization.

Lemma 3.2. Let R be a finite commutative ring with nonzero identity such that its all maximal ideals have size ≤ 6. Then the following statements hold true:

(a) If R is a local ring, then it is either a field or is isomorphic to one of the rings given by Table 1.

(b) If R is a product of two local rings, then it is isomorphic to one of the rings given by Table 2.

(c) If R is a product of three local rings, then it is isomorphic to one of ℤ_{2}×ℤ_{2}×ℤ_{2} and ℤ_{2} × ℤ_{2} × ℤ_{3}.

Proof. (a) This part of the lemma may be obtained by using [6] together with some known results on the structures of small local rings.

(b) By the assumption, R is a product of two local rings, say R_{1} and R_{2}. Let 𝔪_{1} and 𝔪_{2} be the maximal ideals of R_{1} and R_{2}, respectively.

\begin{matrix}|𝔪| & |R| & R \\ 5 & 25 & ℤ_{25}, \frac{ℤ_{5}[x]}{(x^{2})} \\ 4 & 16 & \frac{ℤ_{4}[x]}{(x^{2}+x+1)}, \frac{𝔽_{4}[x]}{(x^{2})} \\ 4 & 8 & ℤ_{8}, \frac{ℤ_{2}[x]}{(x^{3})}, \frac{ℤ_{4}[x]}{(2x,x^{2})}, \frac{ℤ_{4}[x]}{(2x,x^{2}−2)}, \frac{ℤ_{2}[x,y]}{(x,y)^{2}} \\ 3 & 9 & ℤ_{9}, \frac{ℤ_{3}[x]}{(x^{2})} \\ 2 & 4 & ℤ_{4}, \frac{ℤ_{2}[x]}{(x^{2})} \end{matrix}

Table 1. Finite commutative local rings R with maximal ideals 𝔪 of size ≤ 6.

\begin{matrix}|𝔪| & R \\ ≤ 6 & ℤ_{3} × ℤ_{4}, ℤ_{3} × \frac{ℤ_{2}[x]}{(x^{2})} \\ ≤ 5 & ℤ_{2} × ℤ_{5}, ℤ_{3} × ℤ_{5}, 𝔽_{4} × ℤ_{5}, ℤ_{5} × ℤ_{5} \\ ≤ 4 & ℤ_{2} × ℤ_{4}, ℤ_{2} × 𝔽_{4}, ℤ_{2} × \frac{ℤ_{2}[x]}{(x^{2})}, ℤ_{3} × 𝔽_{4}, 𝔽_{4} × 𝔽_{4} \\ ≤ 3 & ℤ_{2} × ℤ_{3}, ℤ_{3} × ℤ_{3} \\ ≤ 2 & ℤ_{2} × ℤ_{2} \end{matrix}

Table 2. Finite commutative rings R with maximal ideals 𝔪 of size ≤ 6 which are a product of two local rings.

First, we may assume that none of R_{1} and R_{2} is a field. In this case, for i = 1, 2, we have |R_{i}| ≥ 4 and |𝔪_{i}| ≥ 2. Note that 𝔪_{1} × R_{2} is a maximal ideal of R with |𝔪_{1} × R_{2}| ≥ 8, a contradiction. Hence, this is not the case and so either both of R_{1} and R_{2} are fields or only one of them is a field.

Case 1: Both of R_{1} and R_{2} are fields. In this case, {0} × R_{2} and R_{1} × {0} are the maximal ideals of R. Now, by the assumption, for i = 1, 2, we obtain that |R_{i}| ≤ 5. Therefore, R is isomorphic to one of the following rings:

ℤ_{2} × ℤ_{2}, ℤ_{2} × ℤ_{3}, ℤ_{2} × 𝔽_{4}, ℤ_{2} × ℤ_{5},

ℤ_{3} × ℤ_{3}, ℤ_{3} × 𝔽_{4}, ℤ_{3} × ℤ_{5},
𝔽_{4} × 𝔽_{4}, 𝔽_{4} × ℤ_{5},
ℤ_{5} × ℤ_{5}.

Case 2: Only, say R_{1}, is a field. In this case, R_{2} is not a field and so we have |R_{2}| ≥ 4 and |𝔪_{2}| ≥ 2. Note that {0} × R_{2} and R_{1} × 𝔪_{2} are the maximal ideals of R.

Now, by the assumption, |R_{1}| ≤ 3 and 4 ≤ |R_{2}| ≤ 6. Therefore, R is isomorphic to one of ℤ_{2} × ℤ_{4}, ℤ_{2} × \frac{ℤ_{2}[x]}{(x^{2})}, ℤ_{3} × ℤ_{4} or ℤ_{3} × \frac{ℤ_{2}[x]}{(x^{2})}.

(c) By the assumption, R is a product of three local rings, say R_{1}, R_{2} and R_{3}. If one of the R_{i}s has size ≥ 4, then R has a maximal ideal of size ≥ 8, a contradiction. If two of the R_{i}s have size equal to 3, then R has a maximal ideal of size ≥ 9, again a contradiction. Therefore, R is isomorphic to one of ℤ_{2} ×ℤ_{2} ×ℤ_{2} and ℤ_{2} ×ℤ_{2} ×ℤ_{3}. □

Let us continue the process of classification by looking at some projective Jacobson graphs.

Lemma 3.3. If R is one of the rings ℤ_{2} × ℤ_{5}, ℤ_{3} × ℤ_{4}, ℤ_{3} × 𝔽_{4} and ℤ_{3} × \frac{ℤ_{2}[x]}{(x^{2})}, then γ̄(𝔍_{R}) = 1.

Proof. By Proposition 2.2, we have γ̄(𝔍_{R}) ≥ 1. On the other hand, for each of the given rings R in the statement of the lemma, one of the Figures 2, 3, 4 and 5 gives an embedding of the Jacobson graph 𝔍_{R} on a projective plane. Hence, γ̄(𝔍_{R}) = 1. □

[figure: 14; 04; 13; 10; 11; 02; 03; 01; 12]

Figure 2. Embedding of the Jacobson graph of ℤ_{2} × ℤ_{5} on a projective plane.

Figure 3. Embedding of the Jacobson graph of ℤ_{3} × ℤ_{4} on a projective plane.

[figure: 03; 01; 23; 21; 20; 22; 12; 10; 13; 11; 23; 03; 01]

[figure: 1(1 + x); 2x; 10; 20; 1x; 11; 21; 2(1 + x); 01; 0(1 + x); 0x]

Figure 4. Embedding of the Jacobson graph of ℤ_{3} × 𝔽_{4} on a projective plane.

[figure: 0(1 + x); 01; 1(1 + x); 11; 10; 1x; 2x; 20; 21; 2(1 + x); 1(1 + x); 0(1 + x); 01]

Figure 5. Embedding of the Jacobson graph of ℤ_{3} × \frac{ℤ_{2}[x]}{(x^{2})} on a projective plane.

We continue the paper by looking at some rings whose Jacobson graphs have nonorientable genus greater than one. Let us start with the following lemma.

Lemma 3.4. If R is one of the rings \frac{ℤ_{4}[x]}{(x^{2}+x+1)} and \frac{𝔽_{4}[x]}{(x^{2})}, then γ̄(𝔍_{R}) = 2.

Proof. By Proposition 2.3, the Jacobson graph 𝔍_{R} is isomorphic to the disjoint union of a copy of K_{4} and a copy of K_{4,4}. Hence, by Propositions 2.8, 2.4 and 2.5, we obtain that γ̄(𝔍_{R}) = 2. □

Lemma 3.5. If R is one of the rings ℤ_{25} and \frac{ℤ_{5}[x]}{(x^{2})}, then γ̄(𝔍_{R}) = 7.

Proof. By Proposition 2.3, the Jacobson graph 𝔍_{R} is isomorphic to the disjoint union of two copies of K_{5} and a copy of K_{5,5}. Hence, by Propositions 2.8, 2.4 and 2.5, we obtain that γ̄(𝔍_{R}) = 7. □

Before stating the next lemma, we recall that for a graph G and a subset V of the vertex set of G, the subgraph of G whose vertex set is V and whose edge set consists of all edges in G that have both endpoints in V is called the induced subgraph of G by V and is denoted by ⟨V ⟩.

Lemma 3.6. If R = ℤ_{3} × ℤ_{5}, then γ̄(𝔍_{R}) ≥ 2.

Proof. Consider the following two subsets V_{1} and V_{2} of the vertex set of 𝔍_{R}:

V_{1} = {(1, 0), (1, 1), (1, 2), (1, 3), (1, 4)},
V_{2} = {(2, 0), (2, 1), (2, 2), (2, 3), (2, 4)}.

It is easy to see that the two subgraphs ⟨V_{1}⟩ and ⟨V_{2}⟩ are disjoint, and ⟨V_{1}⟩ ≅ ⟨V_{2}⟩ ≅ K_{5}. Therefore, it follows from Propositions 2.8 and 2.4 that γ̄(𝔍_{R}) ≥ 2. □

Lemma 3.7. If R = 𝔽_{4} × 𝔽_{4}, then γ̄(𝔍_{R}) ≥ 3.

Proof. Let G be a graph formed by removing the following 8 edges from the Jacobson graph 𝔍_{R}:

{(0, 1), (1, 1)}, {(1, 1), (1, 0)},
{(1, x), (1, x + 1)}, {(x, 1), (x + 1, 1)},
{(x, x), (x + 1, x)}, {(x + 1, x), (x + 1, x + 1)},
{(x + 1, x + 1), (x, x + 1)}, {(x, x + 1), (x, x)}.

It is easy to see that G is a connected graph with 15 vertices and 32 edges and has no triangles, so by Proposition 2.7, we obtain that γ̄(G) ≥ 3. This implies that γ̄(𝔍_{R}) ≥ 3. □

Lemma 3.8. If R = ℤ_{2} × ℤ_{2} × ℤ_{3}, then γ̄(𝔍_{R}) ≥ 3.

Proof. Since ℤ_{2} ×ℤ_{2} ×ℤ_{3} ≅ ℤ_{2} ×ℤ_{6}, it is enough to prove the lemma for R = ℤ_{2} ×ℤ_{6}. It is easy to see that 𝔍_{R} is a connected graph with 11 vertices and 35 edges, so by Proposition 2.6, we obtain that γ̄(𝔍_{R}) ≥ \frac{35}{3} − 11 + 2 > 2. Therefore, γ̄(𝔍_{R}) ≥ 3. □

Lemma 3.9. If R is one of the rings 𝔽_{4} × ℤ_{5} and ℤ_{5} × ℤ_{5}, then γ̄(𝔍_{R}) ≥ 5.

Proof. Consider the following two subsets V_{1} and V_{2} of the vertex set of 𝔍_{R}. If R = 𝔽_{4} × ℤ_{5} is the case, we may consider

V_{1} = {(x, 0), (x, 1), (x, 2), (x, 3), (x, 4)},
V_{2} = {(x + 1, 0), (x + 1, 1), (x + 1, 2), (x + 1, 3), (x + 1, 4)},

and if R = ℤ_{5} × ℤ_{5} is the case, we may consider

V_{1} = {(2, 0), (2, 1), (2, 2), (2, 3), (2, 4)},
V_{2} = {(3, 0), (3, 1), (3, 2), (3, 3), (3, 4)}.

In each cases, it is easy to see that ⟨V_{1} ∪ V_{2}⟩ has a subgraph isomorphic to K_{5,5}. Therefore, it follows from Propositions 2.8 and 2.5 that γ̄(𝔍_{R}) ≥ 5. □

Now, by using Lemma 3.2 together Proposition 2.2 and all of the remaining lemmas (except Lemma 3.1) one has Table 3.

R  γ̄(𝔍_{R})  type of R
ℤ_{4}, ℤ_{8}, ℤ_{9}, \frac{ℤ_{2}[x]}{(x^{2})}, \frac{ℤ_{2}[x]}{(x^{3})}, \frac{ℤ_{3}[x]}{(x^{2})},  0  a local ring

\frac{ℤ_{4}[x]}{(2x,x^{2})}, \frac{ℤ_{4}[x]}{(2x,x^{2}−2)}, \frac{ℤ_{2}[x,y]}{(x,y)^{2}}, a field

ℤ_{2} × ℤ_{2}, ℤ_{2} × ℤ_{3}, ℤ_{2} × ℤ_{4},  0  a product of two local rings

ℤ_{2} × 𝔽_{4}, ℤ_{2} × \frac{ℤ_{2}[x]}{(x^{2})}, ℤ_{3} × ℤ_{3}

ℤ_{2} × ℤ_{2} × ℤ_{2}  0 a product of three local rings
ℤ_{2} × ℤ_{5}, ℤ_{3} × ℤ_{4},  1  a product of two local rings
ℤ_{3} × 𝔽_{4}, ℤ_{3} × \frac{ℤ_{2}[x]}{(x^{2})}
\frac{ℤ_{4}[x]}{(x^{2}+x+1)}, \frac{𝔽_{4}[x]}{(x^{2})}  2  a local ring
ℤ_{25}, \frac{ℤ_{5}[x]}{(x^{2})}  7  a local ring
ℤ_{3} × ℤ_{5}  ≥ 2 a product of two local rings
𝔽_{4} × 𝔽_{4}  ≥ 3 a product of two local rings
ℤ_{2} × ℤ_{2} × ℤ_{3}  ≥ 3 a product of three local rings
𝔽_{4} × ℤ_{5}, ℤ_{5} × ℤ_{5}  ≥ 5 a product of two local rings

Table 3. Nonorientable genus of all of the Jacobson graphs arising from finite commutative rings R with nonzero identity which are either a local ring, a product of two local rings or a product of three local rings, and all of their maximal ideals have size ≤ 6.

We are now in the position to complete the proof of the Main Theorem.

Proof of the Main Theorem. (⇒): Suppose that R is a finite commutative ring with nonzero identity such that its Jacobson graph is projective, that is, γ̄(𝔍_{R}) = 1. By [9, Page 95], we may write R ≅ R_{1} × ⋯ × R_{ℓ}, where every R_{i} is a local ring with maximal ideal 𝔪_{i}. Since γ̄(𝔍_{R}) = 1, by Lemma 3.1, the size of all of the maximal ideals of R is at most 6. This forces that ℓ ≤ 3, that is, either R is a local ring, R is a product of two local rings, or a product of three local rings. Now, Table 3 implies that R is isomorphic to one of ℤ_{2} × ℤ_{5}, ℤ_{3} × ℤ_{4}, ℤ_{3} × 𝔽_{4} or ℤ_{3} × \frac{ℤ_{2}[x]}{(x^{2})}.

(⇐): If R is isomorphic to one of ℤ_{2} × ℤ_{5}, ℤ_{3} × ℤ_{4}, ℤ_{3} × 𝔽_{4} or ℤ_{3} × \frac{ℤ_{2}[x]}{(x^{2})}, then again by Table 3 we obtain that γ̄(𝔍_{R}) = 1. □

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H. Amraei, Mathematics Section, Department of Basic Sciences, Shahid Rajaee Teacher Training University, P.O. Box 16785-163, Tehran, Iran.

E-mail address: h.amraei@srttu.edu

H. R. Maimani, Mathematics Section, Department of Basic Sciences, Shahid Rajaee Teacher Training University, P.O. Box 16785-163, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

E-mail address: maimani@ipm.ir

M. R. Pournaki, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

E-mail address: pournaki@ipm.ir
URL: http://math.ipm.ac.ir/pournaki/

A. Zaeembashi, Mathematics Section, Department of Basic Sciences, Shahid Rajaee Teacher Training University, P.O. Box 16785-163, Tehran, Iran.

E-mail address: azaeembashi@srttu.edu