ZERO-DIVISOR GRAPHS OF MATRICES OVER COMMUTATIVE RINGS

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1 Communications in Algebra, 37: , 2009 Copyright Taylor & Francis Group, LLC ISSN: print/ online DOI: / ZERO-DIVISOR GRAPHS OF MATRICES OVER COMMUTATIVE RINGS Ivana Božić 1 and Zoran Petrović 2 1 Department of Mathematics, Harvard University, Cambridge, Massachusetts, USA 2 Faculty of Mathematics, University of Belgrade, Belgrade, Serbia Key Words: We investigate the properties of (directed) zero-divisor graphs of matrix rings. Then we use these results to discuss the relation between the diameter of the zero-divisor graph of a commutative ring R and that of the matrix ring M n R. Commutative rings; Matrix rings; Zero-divisor graph Mathematics Subject Classification: Primary 16S50; Secondary 13A99, 05C INTRODUCTION The concept of a zero-divisor graph of a commutative ring was first introduced in Beck (1988), and later redefined in Anderson and Livingston (1999). Redmond (2002) further extended this concept to the noncommutative case, introducing several definitions of a zero-divisor graph of a noncommutative ring. Recently, the diameter and girth of polynomial and power series rings over a commutative ring were studied in Axtell et al. (2005), Anderson and Mulay (2007), and Lucas (2006). Let R be a commutative ring with 1, and let M n R denote the ring of n n matrices over R. Let R and M n R denote the zero-divisor graphs of R and M n R, respectively. The object of this article is to find a relation between the diameters of R and M n R. We view this problem as a natural continuation of the investigation into relations between diameters of zero-divisor graphs of a commutative ring R and polynomials and power series over the same ring. Unlike these results, our case involves an extension of a commutative ring into a noncommutative one and, therefore, relates graphs of commutative and noncommutative rings. In Section 2, we recollect the main definitions and the results which we need for both the commutative and noncommutative case. In Section 3, we prove several theorems about zero-divisor graphs of matrix rings. In Section 4, we turn to our main problem the investigation of possible diameters of M n R in terms of the diameter of R. Received August 20, 2007; Revised February 25, Communicated by I. Swanson. Address correspondence to Zoran Petrović, Faculty of Mathematics, University of Belgrade, Studentski trg 16, Belgrade 11000, Serbia; zoranp@matf.bg.ac.yu 1186

2 2. ZERO-DIVISOR GRAPHS ZERO-DIVISOR GRAPHS OF MATRICES 1187 In this section we recall the definitions and basic properties of zero-divisor graphs of commutative and noncommutative rings. If R is an arbitrary ring, let Z R denote the set of zero-divisors of R, and let Z R denote the set of nonzero zero-divisors of R. If the ring R is noncommutative, we let Z L R and Z R R denote the sets of left and right zero-divisors of R, respectively. For a commutative ring R, we consider the undirected graph R with vertices in the set Z R, such that for distinct vertices a and b there is an edge connecting them if and only if ab = 0. We recall that a graph is connected if there exists a path connecting any two distinct vertices. The distance between two distinct vertices a and b, denoted d a b, is the length of the shortest path connecting them (if such a path does not exist, then d a b = ). The diameter of a graph, denoted diam, is equal to sup d a b a b distinct vertices of The girth of a graph, denoted g, is the length of the shortest cycle in. If R is a noncommutative ring, we define a directed zero-divisor graph R in a similar way (this definition was introduced by Redmond, 2002). A directed graph is connected if there exists a directed path connecting any two distinct vertices. The distance and the diameter are defined in a similar way as well, having in mind that all paths in question are directed. Redmond (2002) also defined an undirected zero-divisor graph of a noncommutative ring R, denoted by R, with vertices in the set Z R and such that for distinct vertices a and b there is an edge connecting them if and only if ab = 0 or ba = 0. The results from the following theorem were proved in Anderson and Livingston (1999) and Mulay (2002). Theorem 2.1. Let R be a commutative ring, with Z R. Then R is always connected, diam R 3 and g R 4. The following two theorems about zero-divisor graphs of noncommutative rings were proved by Redmond (2002). Theorem 2.2. Let R be a noncommutative ring, with Z R. Then R is connected if and only if Z L R = Z R R. If R is connected, then diam R 3. Theorem 2.3. Let R be a noncommutative ring, with Z R. Then R is connected, diam R 3 and g R ZERO-DIVISOR GRAPHS OF M n R In what follows, we always assume that our commutative rings have identity. Our goal is to give a characterization of the possible diameters of M n R in terms of the diameter of R. In this section, we prove theorems which will enable us to

3 1188 BOŽIĆ AND PETROVIĆ give such a characterization. We know that, even if R does not contain zero-divisors, M n R does. Also, we note that there always exist distinct A B Z M n R such that AB is not equal to zero. Thus, diam M n R 2 for all commutative rings R. Unlike the zero-divisor graphs of commutative rings, (directed) zero-divisor graphs of noncommutative rings need not be connected. We proceed to determine whether zero-divisor graphs of matrices over commutative rings are connected. Theorem 3.1. Let R be a commutative ring. Then M n R is connected and diam M n R 3. Proof. We have by Theorem 2.2 that a (directed) zero-divisor graph of a noncommutative ring R is connected if and only if Z L R = Z R R. By the same theorem, if such a graph is connected, then diam R 3. We complete the proof by recalling that a matrix A is either a left or a right zero-divisor in M n R if and only if det A Z R (cf. Theorem 9.1 from Brown, 1993). Our next aim is to establish a relationship between the diameters of R and M n R. Proposition 3.1. Let R be a commutative ring such that Z R. Then diam R diam M n R. Proof. We know that diam R 3 (cf. Theorem 2.1). If diam R = 1 or diam R = 2, then diam M n R 2 diam R. If diam R = 3, then there are different nonzero elements a b c d in R such that ab = 0, bc = 0, cd = 0, and a b c d is the shortest path from a to d in R. Then the matrices A = ai n and D = di n belong to Z M n R, and obviously AD 0. Let us assume that there exists a matrix C = c ij Z M n R such that AC = CD = 0. We then have AC = ac = 0 and CD = dc = 0. Thus, for all i j = 1 n ac ij = dc ij = 0, and since only zero annihilates both a and d, we have c ij = 0 for all i j = 1 n, so C = 0, a contradiction. We have proved that diam M n R 3 = diam R. Let T R be the total quotient ring of a commutative ring R. Anderson et al. (2003) showed that R and T R were isomorphic as graphs, and as a consequence it followed that these two graphs had the same diameter and girth. We will show that the same claim holds for M n R and M n T R as well. We recall that two graphs G and G are isomorphic if there is a bijection G G of vertices such that vertices x and y are adjacent in G if and only if x and y are adjacent in G. Theorem 3.2. Let R be a commutative ring with total quotient ring T R. Then M n R M n T R. Proof. We will construct a proof similar to that of Theorem 2.2 from Anderson et al. (2003). Let R be a commutative ring, and let A M n R. Let ann R A L = X M n R XA = 0 and ann R A R = X M n R AX = 0. For A B M n R, we define A B if and only if ann R A L = ann R B L and ann R A R = ann R B R.

4 ZERO-DIVISOR GRAPHS OF MATRICES 1189 The relation is an equivalence relation on M n R and restricts to an equivalence relation on Z M n R, which is the vertex set of M n R. Let T = T R. We denote the equivalence relations defined as above on Z M n R and Z M n T by R and T, and their respective equivalence classes by A R and A T. If A = a ij M n R and s R Z R, we denote by A/s the matrix a ij /s M n T. Note that A Z M n T if and only if A = B/s for some B Z M n R and s R Z R. We have that Z M n R = A R and Z M n T = A /1 T. Note that there is a bijection between sets of equivalence classes of R and T, given by A R A /1 T, and therefore we can use the same index set. We also note that both unions are disjoint. One can prove that A R = A/1 T for all A Z M n R, and the proof of this fact is completely analogous to the proof of the corresponding fact from Anderson et al. (2003). Therefore, there is a bijection A R A /1 T for each. We define Z M n R Z M n T by X = X if X A R. The map is a bijection from M n R to M n T. We show that XY = 0 in Z M n R if and only if X Y = 0 in Z M n T. Let X A R, Y B R, W A/1 T, Z B/1 T. Note that ann T X R = ann T A R = ann T W R and ann T Y L = ann T B L = ann T Z L. Thus XY = 0 Y ann T X R = ann T W R WY = 0 W ann T Y L = ann T Z L WZ = 0. We conclude that M n R M n T. Corollary 3.1. Let R be a commutative ring with total quotient ring T R. Then diam M n R = diam M n T R. Proof. Since the diameter is a graph invariant the result follows from the previous theorem. We conclude this section by giving an elementary result about the girth of M n R. Proposition 3.2. Let R be a commutative ring. Then g M n R = 3. Proof. Let A = B = and C =

5 1190 BOŽIĆ AND PETROVIĆ A B, and C are distinct nonzero matrices from Z M n R such that AB = BC = CA = DIAMETERS OF R AND M n R In this section we apply the results we have obtained about zero-divisor graphs of matrix rings in order to characterize diam M n R in terms of diam R. First we prove the following two lemmas. Lemma 4.1. Let R be a commutative ring and A a matrix from Z M n R. If all elements of R are either invertible or zero-divisors, then there exists an invertible matrix P M n R such that the first row of the matrix PA (alternatively, the last column of the matrix AP) consists entirely of zero-divisors. Proof. Assume that every row of the matrix A contains an invertible element; otherwise, we can change places of the row which contains only zero-divisors and the first one. Let x n be an invertible element in the last row of matrix A. Multiplying the last row by a suitable element of R, and adding it to all other rows, we can transform A into a matrix A 1 which has all zeros above x n in the last row. If A 1 has a row which contains only zero-divisors, we proceed as above. Otherwise, we let x n 1 be an invertible element in the penultimate row. Similarly, we can transform A 1 into a matrix A 2 which has all zeros above x n and x n 1. Continuing this process, we will either get a matrix A k which has the first row containing only zero-divisors, or we will get a matrix A n 1 which has all zeros in the first row, except possibly for one element x 1. If x 1 is not a zero-divisor, it must be invertible, so A n must also be invertible. This is a contradiction. Therefore, x 1 is a zero-divisor, and the invertible matrix P such that A n 1 = PA is the required one. Lemma 4.2. Let R be a commutative ring. If every finite set of zero-divisors from R has a nonzero annihilator, then diam M n R = 2. Proof. We note that the total quotient ring T = T R of R contains only elements which are either invertible or zero-divisors. In addition, every finite set of zero-divisors from T must have a nonzero annihilator. Let A B Z M n T. By Lemma 4.1, there exist invertible matrices P Q M n T such that the last column of the matrix AP and the first row of the matrix QB consist entirely of zero-divisors. Let S be the set of zero-divisors from the last column of the matrix AP and the first row of the matrix QB. Let a be nonzero annihilator of S, and let C = P Q a 0 0 Then C 0 and AC = CB = 0. Thus diam M n T = 2 and by Corollary 3.1 diam M n R = 2.

6 ZERO-DIVISOR GRAPHS OF MATRICES 1191 Corollary 4.1. If F is a field, then diam M n F = 2. The result of this corollary may also be found in Wu (2005). We can further extend this to integral domains. Proposition 4.1. Let R be an integral domain. Then diam M n R = 2. Proof. Let T denote the quotient field of R. By Corollaries 3.1 and 4.1, we have that diam M n R = diam M n T = 2. We proceed to characterize the diameter of M n R for rings with complete zero-divisor graphs. Theorem 4.1. Let R be a commutative ring such that R 2 2, and let diam R = 1. Then diam M n R = 2. Proof. Since R 2 2, we have by Theorem 2.8 in Anderson and Livingston (1999) that xy must be zero for all x y Z R. Therefore, any finite set of zero-divisors in R has a nonzero annihilator (any nonzero zero-divisor will do). Thus, by Lemma 4.2 we have diam M n R = 2. We show in Theorem 4.2 that diam M n 2 2 = 3. By Theorem 2.7 of Anderson and Mulay (2007), if R is a commutative ring and diam R = 2, then exactly one of the following holds: (1) Z R is a prime ideal in R; or (2) T R = K 1 K 2, where both K i are fields. We proceed to examine the diameter of M n R in each of the two cases. Theorem 4.2. Let R be a commutative ring such that T R = K 1 K 2, where both K i are fields. Then diam M n R = 3. Proof. First note that M n K 1 K 2 M n K 1 M n K 2. Also, A 1 A 2 is a zero-divisor in Z M n K 1 M n K 2 if and only if A 1 Z M n K 1 or A 2 Z M n K 2. Let A = A 1 A 2 and B = B 1 B 2 be matrices from M n K 1 K 2, where A 1 and B 2 are the identity matrices, while A 2 and B 1 are matrices whose only nonzero entry is 1, which appears in the upper left corner (one uses notation E 11 to denote such a matrix). Both A and B are zero-divisors in the ring M n K 1 M n K 2 and AB 0. We also note that A 1 and B 2 are invertible, and that A 2 Z M n K 2, B 1 Z M n K 1. If there exists a C = C 1 C 2 Z M n K 1 M n K 2 such that AC = 0 and CB = 0, then it must be C 1 = 0 and C 2 = 0. Hence, diam M n T 3 and thus diam M n T = 3. By Corollary 3.1 we conclude that diam M n R = 3. In the remaining case, when Z R is an ideal in R, we prove the following result about the diameter of M n R for Noetherian rings R. Theorem 4.3. Let R be a Noetherian ring such that diam R = 2. If Z R is a prime ideal in R, then diam M n R = 2.

7 1192 BOŽIĆ AND PETROVIĆ Proof. We have by Theorem 82 in Kaplansky (1974) that Z R is annihilated by a single element, say a. By Lemma 4.2, we have diam M n R = 2. However, if diam R = 2 and Z R is a prime ideal, R being Noetherian is not a necessary condition for diam M n R to be 2. We recall that R is a McCoy ring if every finitely generated ideal in R contained in Z R has a nonzero annihilator. An example of a McCoy ring which is not Noetherian is a polynomial ring R X 1 X 2, where the ring R is any commutative ring with identity. This follows easily from the fact that R X is always a McCoy ring (see Huckaba and Keller, 1979). Proposition 4.2. If R is a McCoy ring such that diam R = 2 and Z R is a prime ideal, then diam M n R = 2. Proof. Let S be a finite set of zero-divisors from R, and let I be the ideal generated by the elements of S. Since Z R is an ideal, I Z R, and thus I has a nonzero annihilator. By Lemma 4.2, we have that diam M n R = 2. When R is a commutative ring such that diam R = 2, Z R is a prime ideal, and R is not McCoy, the question of whether diam( M n R = 2 remains open. ACKNOWLEDGMENTS The authors would like to thank the anonymous referee for his/her helpful comments which have significantly improved the presentation of the results in this article. This work was partially supported by the Ministry of Science and Environmental Protection of the Republic of Serbia Project # REFERENCES Anderson, D. F., Livingston, P. S. (1999). The zero-divisor graph of a commutative ring. J. Algebra 217: Anderson, D. F., Mulay, S. B. (2007). On the diameter and girth of a zero-divisor graph. J. Pure App. Algebra 210(2): Anderson, D. F., Levy, R., Shapiro, J. (2003). Zero-divisor graphs, von Neumann regular rings and Boolean algebras. J. Pure Appl. Algebra 180: Axtell, M., Coykendall, J., Stickles, J. (2005). Zero-divisor graphs of polynomial and power series over commutative rings. Comm. Alg. 33: Beck, I. (1988). Coloring of commutative rings. J. Algebra 116: Brown, W. (1993). Matrices Over Commutative Rings. Marcel Dekker, Inc. Huckaba, J. A., Keller, J. M. (1979). Annihilation of ideals in commutative rings. Pac. J. Math. 83: Kaplansky, I. (1974). Commutative Rings. Chicago: The University of Chicago Press. Lucas, T. G. (2006). The diameter of a zero divisor graph. J. Algebra 301: Mulay, S. B. (2002). Cycles and symmetries of zero-divisors. Comm. Alg. 30(7): Redmond, S. (2002). The zero-divisor graph of a noncommutative ring. International J. Commutative Rings 1(4): Wu, T. (2005). On directed zero-divisor graphs of finite rings. Discrete Math. 296:73 86.