Acta Math. Hungar. Acta Math. Hungar. https://doi.org/10.1007/s10474-018-0815-x
DOI: 1 0
THE TRACE GRAPH OF THE MATRIX RING OVER A FINITE COMMUTATIVE RING F. A. A. ALMAHDI1,∗ , K. LOUARTITI2 and M. TAMEKKANTE3 1
Department of Mathematics, Faculty of Sciences, King Khalid University, P. O. Box 9004, Abha, Saudi Arabia e-mail:
[email protected] 2
3
Department of Mathematics, Faculty of Science, Ben M’Sik, Box 7955, Sidi Othmane, University Hassan II, Casablanca, Morocco e-mail:
[email protected]
Laboratory MACS, Faculty of Sciences Zitoune, University Moulay Ismail, 5000, P. B. 11201, Meknes, Morocco e-mail:
[email protected] (Received January 24, 2018; revised February 3, 2018; accepted February 9, 2018)
Abstract. Let R be a commutative ring and let n > 1 be an integer. We introduce a simple graph, denoted by Γt (Mn (R)), which we call the trace graph of the matrix ring Mn (R), such that its vertex set is Mn (R)∗ and such that two distinct vertices A and B are joined by an edge if and only if Tr(AB) = 0 where Tr(AB) denotes the trace of the matrix AB. We prove that Γt (Mn (R)) is connected with diam(Γt (Mn (R))) = 2 and gr(Γt (Mn (R))) = 3. We investigate also the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γt (Mn (R)). Hence, we use the notion of the irregularity index of a graph to characterize rings with exactly one nontrivial ideal.
1. Introduction Throughout the paper R will denote a commutative ring with identity 1 �= 0. If X is either an element or a subset of the ring R, then annR (X) denotes the annihilator of X in R. If X is anysubset of a ring, then X ∗ = X\{0}. The concept of a zero-divisor graph was first introduced by Beck in 1988 for his study of the coloring of a commutative ring [8]. In his work, all elements of the ring were vertices of the graph. In [6], Anderson and Livingston introduced and studied the zero-divisor graph whose vertices are the non-zero zero-divisors. Let R be a ring and let Z(R) denote the set of ∗ Corresponding
author. Key words and phrases: zero-divisor, matrix ring, zero-divisor graph, trace graph. Mathematics Subject Classification: 16S50, 13A99, 05C99.
c 2018 0236-5294/$ 20.00 © � 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akadémiai Kiadó, Budapest, Hungary
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F. F. A. A. A. A. ALMAHDI, ALMAHDI, K. K. LOUARTITI LOUARTITI and and M. M. TAMEKKANTE TAMEKKANTE
zero-divisors of R. The zero-divisor graph of a ring R, denoted by Γ(R), is the simple graph whose vertices are the elements of the set Z(R)∗ and, for distinct x, y ∈ Z(R)∗ , there is an edge connecting x and y if and only if xy = 0. Let Γ be a graph. We say that Γ is connected if there is a path between any two distinct vertices of Γ. For distinct vertices x and y of Γ, let d(x, y) be the length of the shortest path between x and y (d(x, y) = ∞ if there is no such path). The diameter of Γ is diam(Γ) := sup{d(x, y) | x and y are distinct vertices of Γ}. The girth of Γ, denoted by gr(Γ), is defined as the length of the shortest cycle in Γ (gr(Γ) = ∞ if Γ contains no cycles). It is proved that Γ(R) is connected with diam(Γ(R)) ≤ 3 ([6, Theorem 2.3]) and gr(Γ(R)) ≤ 4 if Γ(R) contains a cycle ([16, (1.4)]). Thus, diam(Γ(R)) = 0, 1, 2 or 3 and gr(Γ(R)) = 3, 4 or ∞ (examples of different cases can be found in [5]). The zero-divisor graphs of commutative rings have attracted the attention of several researchers (see, for instance, [2–4, 7,10,21]). For a survey and recent results concerning zero-divisor graphs of commutative rings, we refer the reader to [5]. A general reference for graph theory is [22]. Redmond [18] introduced the concept of the zero-divisor graph for noncommutative rings as follows: Let S be a noncommutative ring, and let Z(S) = {x ∈ S | xy = 0 or yx = 0 for some y ∈ S ∗ }. The directed zerodivisor graph of S, denoted by Γ(S), is the simple graph whose vertices are the elements of the set Z(S)∗ and, for distinct x, y ∈ Z(R)∗ , there is a directed edge x → y connecting x and y if and only if xy = 0. Let ZL (S) denote the set of left zero-divisors of S, i.e., ZL (S) = {x ∈ S | xy = 0 for some y ∈ S ∗ }. Similarly, let ZR (S) denote the set of right zero-divisors of S. It is proved (in [18, Theorem 2.3]) that if Z(S)∗ �= ∅ then Γ(S) is connected if and only if ZL (S) = ZR (S). Moreover, if Γ(S) is connected then diam(Γ(S)) ≤ 3. Let n be a positive integer and let Mn (R) denote the ring of n × n matrices over the ring R. In [9], Bozic and Petrovic investigated the zerodivisor graph Γ(Mn (R)). They proved, among other things, that Γ(Mn (R)) is connected, diam(Γ(Mn (R))) = 2 provided R is an integral domain, and if Z(R)∗ �= ∅ then diam(Γ(R) ≤ diam(Γ(Mn (R))) ([9, Theorem 3.1 and Propositions 3.1 and 4.1]). The study of zero-divisor graphs of the ring of matrices motivated many works (see for instance [13,14]). Let R be a (commutative) ring and n be a positive integer, and let Tr(A) denote the trace of the matrix A ∈ Mn (R). In this paper, we introduce a simple graph, denoted by Γt (Mn (R)), which we call the trace graph of the matrix ring Mn (R), such that its vertex set is Vn = A ∈ Mn (R)∗ | ∃B ∈ Mn (R)∗ such that Tr(AB) = 0 , and such that two distinct vertices A and B are joined by an edge if and only if Tr(AB) = 0. Note that, except in the case n = 1, if Tr(A) = 0 then there Acta Hungarica Acta Mathematica Mathematica Hungarica
