RINGS WHICH ARE GENERATED BY THEIR UNITS: A GRAPH THEORETICAL APPROACH

H. R. MAIMANI, M. R. POURNAKI, AND S. YASSEMI

Dedicated to the memory of Maria Silvia Lucido

Abstract. The ring ℤ_{2} × ℤ_{2}, having only one unit, cannot be generated by its units. It turns out, in the general theory of rings, that this is essentially the only example. In this note, we give an elementary proof of “A finite commutative ring with nonzero identity is generated by its units if and only if it cannot have ℤ_{2} × ℤ_{2} as a quotient.” The proof uses graph theory, and offers, as a byproduct, that in this case, every element is the sum of at most three units.

1. Introduction

The study of rings which are generated additively by their units seems to have arisen in 1953–1954 when Wolfson [13] and Zelinsky [14] proved, independently, that if V is a finite or infinite dimensional vector space over a division ring D, then every linear transformation is the sum of two nonsingular linear transformations unless dim V = 1 and D = ℤ_{2}. This implies that the ring of linear transformations End_{D}(V) is generated additively by its units. In fact, every element of End_{D}(V) is the sum of two units except for one obvious case when V is a one dimensional vector space over ℤ_{2}. Wolfson’s and Zelinsky’s result caused quite a bit of interest in the study of rings that are generated by their units.

In 1958, Skornyakov [8, Page 167, Problem 31], posed the problem of determining which regular rings are generated by their units. More precisely, he asked: Is every element of a von Neumann regular ring, which cannot have ℤ_{2} as a quotient, a sum of units? – This question of Skornyakov was answered negatively by Bergman in 1977 (see [5] which is a significant contribution to the theory of von Neumann regular rings). Bergman constructed a von Neumann regular algebra in which not all elements are sums of units.

In 1968, while apparently unaware of Skornyakov’s book, Ehrlich [2] produced a large class of regular rings generated by their units. He proved that if R is a ring such that 2 is a unit and for every a ∈ R there exists a unit u ∈ R such that aua = a, then every element of R is the sum of two units.

2000 Mathematics Subject Classification. Primary: 13M05, 16U60; Secondary: 05C38, 05C40.

Key words and phrases. Unit element, Connectedness, Diameter.

The research of H. R. Maimani and M. R. Pournaki was in part supported by a grant from IPM (No. 87050213 and No. 87200111). The research of S. Yassemi was in part supported by a grant from the University of Tehran (No. 6103023/1/07).

In 1974, Raphael [7] launched a systematic study of rings generated by their units, which he calls S-rings.

Finally, in 1976, Fisher and Snider [3] proved that if R is a von Neumann regular ring with primitive factor rings artinian and 2 is a unit, then every element of R can be expressed as the sum of two units.

In 1998, Wolfson’s and Zelinsky’s result was reproved by Goldsmith, Pabst and Scott where they remarked that this result can hardly be new but they were unable to find any reference to it in the literature (see [4]). Interest in this topic increased recently after they defined the unit sum number in [4].

For additional historical background the reader is referred to the paper [10], which also contains references to recent work in this area. Also see [9] for a survey of rings which are generated by their units.

The purpose of this note is to give an elementary proof of Theorem 1.1. The proof uses graph theory, and offers, as a byproduct, that if R is a finite commutative ring with nonzero identity which is generated by its units, then every element of R can be written as a sum of at most three units.

Theorem 1.1 ([7], Corollary 7). Let R be a finite commutative ring with nonzero identity. Then R is generated by its units if and only if R cannot have ℤ_{2} × ℤ_{2} as a quotient.

2. Basic notation and properties of graphs

In this section we introduce some notation and definitions of graphs that will be used throughout the note. We also state and prove Lemmas 2.1 and 2.2 which are required in Section 3. Here, by a graph G we mean a finite undirected graph without loops or multiple edges (unless otherwise specified). The reader is referred to [1] and [12] for a fuller treatment of the subject.

For a graph G, let V (G) denote the set of vertices. Let G be a graph and suppose x, y ∈ V (G). We recall that a walk between x and y is a sequence x = v_{0}, e_{1}, v_{1}, …, e_{k}, v_{k} = y of vertices and edges of G, denoted by

x = v_{0} ⟶^{e_{1}} v_{1} ⟶ ⋯ ⟶^{e_{k}} v_{k} = y,

such that for every i with 1 ≤ i ≤ k, the edge e_{i} has endpoints v_{i−1} and v_{i}. Also a path between x and y is a walk between x and y without repeated vertices. The number of edges in a walk (counting repeats) or a path is called its length.

For the proof of Theorem 1.1 we need the following well-known fact. We state and prove it here for the convenience of the reader.

Lemma 2.1. Let x and y be distinct vertices of a graph G. If there is a walk between x and y then there is also a path between x and y.

Proof. By assumption there is a walk between x and y and so we may select a walk

W : x = v_{0} ⟶^{e_{1}} v_{1} ⟶ ⋯ ⟶^{e_{k}} v_{k} = y

of minimal length k between x and y. If W is not a path, select a vertex that appears twice, say v_{i} = v_{j} where i < j. Consider

W′ : x = v_{0} ⟶^{e_{1}} v_{1} ⟶ ⋯ ⟶^{e_{i}} v_{i} ⟶^{e_{j+1}} v_{j+1} ⟶ ⋯ ⟶^{e_{k}} v_{k} = y.

Then W′ is a walk between x and y with length shorter than k, a contradiction. Therefore W is a path between x and y. □

A graph G is called connected if for all vertices x and y there exists a path between x and y. Otherwise, G is called disconnected.

A bipartite graph is one whose vertex-set is partitioned into two (not necessarily nonempty) disjoint subsets, called parts, in such a way that the two end vertices for each edge lie in distinct parts. Among bipartite graphs, a complete bipartite graph is one in which each vertex is joined to every vertex that is not in the same part.

Let G_{1} and G_{2} be two vertex-disjoint graphs. The category product of G_{1} and G_{2} is denoted by G_{1} ˙×G_{2}. That is, V (G_{1} ˙×G_{2}) := V (G_{1}) × V (G_{2}); two distinct vertices (x, y) and (x′, y′) are adjacent if and only if x is adjacent to x′ in G_{1} and y is adjacent to y′ in G_{2}.

We now state and prove the following lemma which will be used in the proof of Theorem 1.1. A bipartite graph is nontrivial if both parts of its vertex-set are nonempty. For more information on this lemma we refer the reader to [11].

Lemma 2.2. Let G_{1} and G_{2} be two bipartite graphs at least one of which is nontrivial. Then G_{1} ˙×G_{2} is disconnected.

Proof. We assume that G_{2} is nontrivial. Thus G_{1} is partitioned into two disjoint subsets X_{1} and Y_{1} as well as G_{2} into two disjoint subsets X_{2} and Y_{2} in such a way that |X_{1}| ≥ 1, |X_{2}| ≥ 1 and |Y_{2}| ≥ 1. Choose a ∈ X_{1}, b ∈ X_{2} and c ∈ Y_{2}. We claim that there is no path between (a, b) and (a, c) in G_{1} ˙×G_{2}. In order to do this, suppose in contrary that, there is a path P between (a, b) and (a, c) in G_{1} ˙×G_{2}:

P : (a, b) ⟶^{e_{1}} (a_{1}, b_{1}) ⟶^{e_{2}} (a_{2}, b_{2}) ⟶ ⋯ ⟶^{e_{n−1}} (a_{n−1}, b_{n−1}) ⟶^{e_{n}} (a, c).

We now obtain the walk Ŵ in G_{1} and the walk W̃ in G_{2} both with length n:

Ŵ : a ⟶^{ê_{1}} a_{1} ⟶^{ê_{2}} a_{2} ⟶ ⋯ ⟶^{ê_{n−1}} a_{n−1} ⟶^{ê_{n}} a,

W̃ : b ⟶^{ẽ_{1}} b_{1} ⟶^{ẽ_{2}} b_{2} ⟶ ⋯ ⟶^{ẽ_{n−1}} b_{n−1} ⟶^{ẽ_{n}} c.

The existence of the walk Ŵ implies that n is even while the existence of the walk W̃ implies that n is odd, a contradiction. Thus there is no path between (a, b) and (a, c) in G_{1} ˙×G_{2}, which implies that G_{1} ˙×G_{2} is disconnected. □

Let us consider yet a few more definitions required for a complete understanding of the next section. For a graph G and vertices x and y of G, the distance between x and y, denoted by d(x, y), is the number of edges in a shortest path between x and y. If there is no path between x and y then we write d(x, y) = ∞. We recall that the largest distance among all distances between pairs of the vertices of a graph G is called the diameter of G and is denoted by diam(G). Finally, for a given vertex x ∈ V (G), the neighbor set of x is the set N_{G}(x) := {v ∈ V (G) | v is adjacent to x}. Moreover, if G has a loop at vertex x, then we always assume that x ∈ N_{G}(x).

3. Proof of Theorem 1.1

In this section, using the results presented in Section 2, we are able to prove Theorem 1.1.

Let R be an arbitrary finite associative ring R with nonzero identity, say 1, which is preserved by homomorphisms and inherited by subrings. Let U_{R} be the set of units of R. We attach a graph to R, denoted by G_{R}, based on the elements and units of R. This graph is obtained by letting R be the set of vertices and defining distinct vertices x and y to be adjacent if and only if x + y ∈ U_{R}. If we omit the word “distinct” in the definition of G_{R}, we obtain the graph \overline{G}_{R}; this graph may have loops. Note that if 2 ∉ U_{R}, then \overline{G}_{R} = G_{R}.

The graphs in Fig. 1 are the graphs attached to the rings indicated.

\begin{aligned}[figure: 0; 1; 2; G_{ℤ_{3}}] & [figure: 0; 1; 2; 3; G_{ℤ_{4}}] & [figure: 10; 11; 00; 01; G_{ℤ_{2}×ℤ_{2}}] & [figure: 0; 1; 2; _{\overline{G}_{ℤ_{3}}}] \\ & Figure 1. \end{aligned}

It is easy to see that, for given rings R and S, if R ≅ S as rings, then G_{R} ≅ G_{S} as graphs. Also we have \overline{G}_{R} ˙×\overline{G}_{S} ≅ G_{R×S}.

In Fig. 2 we illustrate these points for the direct product of the rings ℤ_{2} and ℤ_{3}.

\begin{aligned}[figure: 0; 1; 2; _{\overline{G}_{ℤ_{3}}}] & ˙× & [figure: 0; 1; _{\overline{G}_{ℤ_{2}}}; ≅] & [figure: 00; 11; 21; 10; 20; 01; G_{ℤ_{3}×ℤ_{2}}] & ≅ & [figure: 0; 5; 1; 2; 4; 3; G_{ℤ_{6}}] \\ & & Figure 2. \end{aligned}

We need the following result, which is useful in the sequel.

Lemma 3.1. Let R be a finite commutative local ring with maximal ideal 𝔪. Then the following statements hold:

(a) If |R/𝔪| = 2, then G_{R} is a complete bipartite graph.

(b) If |R/𝔪| > 2, then for every x, y ∈ R we have N_{\overline{G}_{R}}(x) ∩ N_{\overline{G}_{R}}(y) ≠ ∅.

Proof. Part (a): Let X = 𝔪 and Y = R \ 𝔪. We have V (G_{R}) = X ∪ Y and X ∩ Y = ∅. Therefore X and Y partition V (G_{R}) into two subsets. It is clear that no pair of distinct elements of X are adjacent. We show that no distinct elements of Y are adjacent. In order to do this, fix an element in R \ 𝔪, say a. By assumption we have R = 𝔪 ∪ (𝔪 + a) = 𝔪 ∪ (𝔪 + (−a)). Now for distinct elements x and y in R \ 𝔪, we may write x = m + a and y = m′ − a where m, m′ ∈ 𝔪. If x + y ∈ U_{R}, then we conclude that m + m′ ∈ U_{R}. Therefore 𝔪 has a unit element and so 𝔪 = R, a contradiction. Thus x + y ∉ U_{R}, which implies that x and y are not adjacent. Therefore no distinct elements of Y are adjacent. Hence G_{R} is a bipartite graph.

Suppose that x ∈ X and y ∈ Y are given. If x + y ∉ U_{R}, then x + y ∈ X and so y ∈ X, a contradiction. Thus x + y ∈ U_{R}, which implies that x and y are adjacent. Therefore each vertex of X is joined to every vertex of Y and so G_{R} is complete bipartite.

Part (b): By assumption we conclude that |U_{R}| ≥ 2|R|/3. Suppose that x is an arbitrary element of R and fix it. There are two possibilities: either 2x ∉ U_{R} or 2x ∈ U_{R}. If 2x ∉ U_{R}, then \overline{G}_{R} has no loop at vertex x. On the other hand, for every element u − x, where u ∈ U_{R}, we have u − x ≠ x and u − x is adjacent to x in \overline{G}_{R}. This implies that \{u − x | u ∈ U_{R}\} ⊆ N_{\overline{G}_{R}}(x) and so |N_{\overline{G}_{R}}(x)| ≥ |U_{R}| ≥ 2|R|/3. If 2x ∈ U_{R}, then \overline{G}_{R} has a loop at vertex x. On the other hand, for every element u − x, where u ∈ U_{R} \\{2x\}, we have u−x ≠ x and u−x is adjacent to x in \overline{G}_{R}. This implies that \{u−x | u ∈ U_{R} \\{2x\}\}∪\{x\} = \{u−x | u ∈ U_{R}\} ⊆ N_{\overline{G}_{R}}(x) and so we have again |N_{\overline{G}_{R}}(x)| ≥ |U_{R}| ≥ 2|R|/3. Therefore, in both cases, we have |N_{\overline{G}_{R}}(x)| ≥ 2|R|/3.

Now, for every x, y ∈ R,

|N_{\overline{G}_{R}}(x) ∩ N_{\overline{G}_{R}}(y)| = |N_{\overline{G}_{R}}(x)| + |N_{\overline{G}_{R}}(y)| − |N_{\overline{G}_{R}}(x) ∪ N_{\overline{G}_{R}}(y)|
≥ (2|R|/3) + (2|R|/3) − |R|
= |R|/3
> 0

and so N_{\overline{G}_{R}}(x) ∩ N_{\overline{G}_{R}}(y) ≠ ∅ as required. □

Now let R be a finite commutative ring with nonzero identity and fix it. We want to prove R is generated by its units if and only if R cannot have ℤ_{2}×ℤ_{2} as a quotient. We start with the proposition below which contains a necessary and sufficient condition for G_{R} to be connected.

Proposition 3.2. G_{R} is connected if and only if R is generated by its units.

Proof. (⟹) Suppose that a ∈ R is written by the sum of some units and b ∈ R is adjacent to a in G_{R}. Therefore a + b ∈ U_{R} and so we may write b = c − a, for some c ∈ U_{R}. Thus b is the sum of some units.

Now suppose that x ∈ R is given. Since G_{R} is connected, there exists a path between x and 1 and, therefore, by the above observation we conclude that x is the sum of some units. This means that R is generated by its units.

(⟸) Suppose that a ∈ R. Since R is generated by its units, we may write a = u_{1} + ⋯ + u_{k}, where u_{i} ∈ U_{R}, 1 ≤ i ≤ k. We now have the walk

0 ⟶^{e_{1}} −u_{1} ⟶^{e_{2}} u_{1} + u_{2} ⟶^{e_{3}} −u_{1} − u_{2} − u_{3} ⟶^{e_{4}} u_{1} + u_{2} + u_{3} + u_{4} ⟶ ⋯ ⟶^{e_{k}}

u_{1} + ⋯ + u_{k} = a

between 0 and a, when k is even and the walk

0 ⟶^{e_{1}} u_{1} ⟶^{e_{2}} −u_{1} − u_{2} ⟶^{e_{3}} u_{1} + u_{2} + u_{3} ⟶^{e_{4}} −u_{1} − u_{2} − u_{3} − u_{4} ⟶ ⋯ ⟶^{e_{k}}

u_{1} + ⋯ + u_{k} = a

between 0 and a, when k is odd.

This implies that for every x, y ∈ R there is a walk W_{1} between x and 0 as well as a walk W_{2} between 0 and y. The walks W_{1} and W_{2} together form a walk W between x and y. By using Lemma 2.1, we conclude that there is also a path P between x and y, which implies the connectedness of G_{R}. □

The following proposition contains another necessary and sufficient condition for G_{R} to be connected.

Proposition 3.3. G_{R} is connected if and only if R cannot have ℤ_{2} ×ℤ_{2} as a quotient.

Proof. Every finite commutative ring with nonzero identity is isomorphic to a direct product of finite local rings (see [6, Page 95]). Therefore, we may write R ≅ R_{1} × ⋯ × R_{k}, where every R_{i} is a local ring with maximal ideal 𝔪_{i}.

(⟹) Suppose by contrary that R has ℤ_{2} × ℤ_{2} as a quotient. This implies that for at least two i, for example i = 1, 2, we have |R_{i}/𝔪_{i}| = 2. Now part (a) of Lemma 3.1 implies that G_{R_{1}} and G_{R_{2}} are both bipartite. Thus by using Lemma 2.2, we conclude that G_{R_{1}} ˙×G_{R_{2}} is disconnected.

On the other hand, by the observation just before Fig. 2, we have

G_{R} ≅ \{ \begin{matrix}\overline{G}_{R_{1}} ˙×\overline{G}_{R_{2}} & if k = 2, \\ (\overline{G}_{R_{1}} ˙×\overline{G}_{R_{2}}) ˙×\overline{G}_{R_{3}×···×R_{k}} & if k ≥ 3. \end{matrix}

But for i = 1, 2 we have 2 ∉ U_{R_{i}} and so \overline{G}_{R_{i}} = G_{R_{i}}. Therefore we obtain

G_{R} ≅ \{ \begin{matrix}G_{R_{1}} ˙×G_{R_{2}} & if k = 2, \\ (G_{R_{1}} ˙×G_{R_{2}}) ˙×\overline{G}_{R_{3}×···×R_{k}} & if k ≥ 3. \end{matrix}

Now the disconnectedness of G_{R_{1}} ˙×G_{R_{2}} implies that G_{R} is also disconnected. This contradiction shows that R cannot have ℤ_{2} × ℤ_{2} as a quotient.

(⟸) By assumption R cannot have ℤ_{2} × ℤ_{2} as a quotient. This implies that for at most one i, we have |R_{i}/𝔪_{i}| = 2. There are the following cases to be considered:

(1) |R_{i}/𝔪_{i}| > 2 holds for every i.

Suppose that x = (x_{1}, …, x_{k}) and y = (y_{1}, …, y_{k}) are arbitrary distinct elements of R_{1} × ⋯ × R_{k}. Since for every i with 1 ≤ i ≤ k we have |R_{i}/𝔪_{i}| > 2, by using part (b) of Lemma 3.1 we conclude that N_{\overline{G}_{R_{i}}}(x_{i}) ∩ N_{\overline{G}_{R_{i}}}(y_{i}) ≠ ∅. Therefore we may choose z_{i} ∈ N_{\overline{G}_{R_{i}}}(x_{i}) ∩ N_{\overline{G}_{R_{i}}}(y_{i}). Thus we have the following walk in \overline{G}_{R_{1}×···×R_{k}}:

(x_{1}, …, x_{k}) ⟶^{e_{1}} (z_{1}, …, z_{k}) ⟶^{e_{2}} (y_{1}, …, y_{k}).

This implies that d(x, y) ≤ 2 and so diam(G_{R}) = diam(G_{R_{1}×···×R_{k}}) ≤ 2.

(2) |R_{i}/𝔪_{i}| > 2 holds for every i except one of them.

First, suppose that k = 1. In this case R ≅ R_{1} is a finite local ring with maximal ideal 𝔪_{1} in such a way |R_{1}/𝔪_{1}| = 2. Thus, if R is a field, then we have R ≅ ℤ_{2} and so diam(G_{R}) = 1. If R is not a field, then by using part (a) of Lemma 3.1 we conclude that G_{R} is complete bipartite with |R| ≥ 4 and so diam(G_{R}) = 2.

Second, suppose that k ≥ 2. In this case we may assume that |R_{1}/𝔪_{1}| = 2 and |R_{i}/𝔪_{i}| > 2 for every i with 2 ≤ i ≤ k. Suppose that x = (x_{1}, x_{2}, …, x_{k}) and y = (y_{1}, y_{2}, …, y_{k}) are arbitrary distinct elements of R_{1} × R_{2} × ⋯ × R_{k}. If either x_{1}, y_{1} ∈ 𝔪_{1} or x_{1}, y_{1} ∉ 𝔪_{1}, then by the same argument as above, we obtain a path between x and y with length at most 2. This implies that d(x, y) ≤ 2. Now, we may assume that x_{1} ∈ 𝔪_{1} and y_{1} ∉ 𝔪_{1}. For every i with 2 ≤ i ≤ k, consider w_{i} as follows:

w_{i} = \{ \begin{matrix}1 & if x_{i} ∈ 𝔪_{i}, \\ 0 & if x_{i} ∉ 𝔪_{i}. \end{matrix}

On the other hand, since for every i with 2 ≤ i ≤ k we have |R_{i}/𝔪_{i}| > 2, by using part (b) of Lemma 3.1 we conclude that N_{\overline{G}_{R_{i}}}(w_{i}) ∩ N_{\overline{G}_{R_{i}}}(y_{i}) ≠ ∅. Therefore we may choose z_{i} ∈ N_{\overline{G}_{R_{i}}}(w_{i}) ∩ N_{\overline{G}_{R_{i}}}(y_{i}). Thus we have the following walk in \overline{G}_{R_{1}×R_{2}×···×R_{k}}:

(x_{1}, x_{2}, …, x_{k}) ⟶^{e_{1}} (y_{1}, w_{2}, …, w_{k}) ⟶^{e_{2}} (x_{1}, z_{2}, …, z_{k}) ⟶^{e_{3}} (y_{1}, y_{2}, …, y_{k}).

This implies that d(x, y) ≤ 3. Therefore, for every distinct x, y ∈ R_{1} × R_{2} × ⋯ × R_{k} we have d(x, y) ≤ 3 and so diam(G_{R}) = diam(G_{R_{1}×R_{2}×···×R_{k}}) ≤ 3.

Therefore in both cases we have diam(G_{R}) ≤ 3. Thus every two vertices of G_{R} are joined by a path with length at most 3, which implies that G_{R} is connected. □

Propositions 3.2 and 3.3 imply that R is generated by its units if and only if R cannot have ℤ_{2} × ℤ_{2} as a quotient which completes the proof of Theorem 1.1. Our proof shows that if R cannot have ℤ_{2}×ℤ_{2} as a quotient, then not only G_{R} is connected, but also diam(G_{R}) ≤ 3. Therefore, we may state

Corollary 3.4. Let R be a finite commutative ring with nonzero identity. If R is generated by its units, or equivalently, R cannot have ℤ_{2} × ℤ_{2} as a quotient, then every element of R can be written as a sum of at most three units.

Acknowledgments

The authors would like to thank the referee for his/her useful suggestions which led to an improvement of the present note. Also thanks are due to S. A. Seyed Fakhari from Sharif University of Technology for his useful comments.

References

[1] J. A. Bondy, U. S. R. Murty, Graph Theory, Springer-Verlag, New York, 2008.

[2] G. Ehrlich, Unit-regular rings, Portugal. Math. 27 (1968), 209–212.

[3] J. W. Fisher, R. L. Snider, Rings generated by their units, J. Algebra 42 (1976), no. 2, 363–368.

[4] B. Goldsmith, S. Pabst, A. Scott, Unit sum numbers of rings and modules, Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 195, 331–344.

[5] D. Handelman, Perspectivity and cancellation in regular rings, J. Algebra 48 (1977), no. 1, 1–16.

[6] B. R. McDonald, Finite Rings with Identity, Pure and Applied Mathematics, vol. 28, Marcel Dekker, Inc., New York, 1974.

[7] R. Raphael, Rings which are generated by their units, J. Algebra 28 (1974), 199–205.

[8] L. A. Skornyakov, Complemented Modular Lattices and Regular Rings, Oliver & Boyd, Edinburgh-London, 1964.

[9] A. K. Srivastava, A survey of rings generated by units, Ann. Fac. Sci. Toulouse Math. (6), to appear.

[10] P. Vámos, 2-Good rings, Q. J. Math. 56 (2005), no. 3, 417–430.

[11] P. M. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc. 13 (1962), 47–52.

[12] D. B. West, Introduction to Graph Theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996.

[13] K. G. Wolfson, An ideal-theoretic characterization of the ring of all linear transformations, Amer. J. Math. 75 (1953), 358–386.

[14] D. Zelinsky, Every linear transformation is a sum of nonsingular ones, Proc. Amer. Math. Soc. 5 (1954), 627–630.

H. R. Maimani, Mathematics Section, Department of Basic Sciences, Shahid Rajaee Teacher Training University, 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/

S. Yassemi, School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran.

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