- Email: [email protected]
The Zero-Divisor Graph of a Commutative Ring
Journal of Algebra 217, 434᎐447 Ž1999. Article ID jabr.1998.7840, available online at http:rrwww.idealibrary.com on
The Zero-Divisor Graph of a Commutative Ring David F. Anderson and Philip S. Livingston Mathematics Department, The Uni¨ ersity of Tennessee, Knox¨ ille, Tennessee 37996 E-mail: [email protected], [email protected] Communicated by Kent R. Fuller Received October 5, 1998
For each commutative ring R we associate a Žsimple. graph ⌫ Ž R .. We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of ⌫ Ž R .. 䊚 1999 Academic Press
1. INTRODUCTION Let R be a commutative ring Žwith 1. and let ZŽ R . be its set of zero-divisors. We associate a Žsimple. graph ⌫ Ž R . to R with vertices ZŽ R .* s ZŽ R . y 04 , the set of nonzero zero-divisors of R, and for distinct x, y g ZŽ R .*, the vertices x and y are adjacent if and only if xy s 0. Thus ⌫ Ž R . is the empty graph if and only if R is an integral domain. The main object of this paper is to study the interplay of ring-theoretic properties of R with graph-theoretic properties of ⌫ Ž R .. This study helps illuminate the structure of ZŽ R .. For x, y g ZŽ R ., define x ; y if xy s 0 or x s y. The relation ; is always reflexive and symmetric, but is usually not transitive. The zero-divisor graph ⌫ Ž R . measures this lack of transitivity in the sense that ; is transitive if and only if ⌫ Ž R . is complete. The idea of a zero-divisor graph of a commutative ring was introduced by I. Beck in w2x, where he was mainly interested in colorings. This investigation of colorings of a commutative ring was then continued by D. D. Anderson and M. Naseer in w1x. Their definition was slightly different than ours; they let all elements of R be vertices and had distinct x and y adjacent if and only if xy s 0. We will denote their zero-divisor graph of R by ⌫0 Ž R .. In ⌫0 Ž R ., the vertex 0 is adjacent to every other vertex, but non-zero-divisors are adjacent only to 0. Note that ⌫ Ž R . is a Žinduced. subgraph of ⌫0 Ž R .. Our results for ⌫ Ž R . have natural analogs to ⌫0 Ž R .; 434 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
ZERO-DIVISOR GRAPH
435
however, we feel that our definition better illustrates the zero-divisor structure of R. In Section 2, we give many examples, we show that ⌫ Ž R . is always connected with diamŽ ⌫ Ž R .. F 3, and we determine when ⌫ Ž R . is a complete graph or a star graph. A key step is characterizing when a vertex is adjacent to every other vertex. In the third section, we study the automorphism group of ⌫ Ž R .. We will include basic definitions from graph theory as needed. Basic references for graph theory are w4, 5, 8x; for commutative ring theory, see w2, 9, 12x. All rings R are commutative with 1 Ž/ 0., and for A ; R, we let A* s A y 04 . As usual, the rings of integers and integers modulo n will be denoted by ⺪ and ⺪ n , respectively, and ⺖q will be the finite field with q elements. To avoid trivialities when ⌫ Ž R . is empty, we will implicitly assume when necessary that R is not an integral domain. 2. PROPERTIES OF ⌫ Ž R . In this section, we show that ⌫ Ž R . is always connected and has small diameter and girth, and we determine which complete graphs and star graphs may be realized as ⌫ Ž R .. We start with some examples which motivate later results Žalso see Examples 2.11 and 2.14.. Easy details Žincluding labeling the vertices. are left to the reader. EXAMPLE 2.1. Ža. Below are the zero-divisor graphs for several rings. Note that these examples show that nonisomorphic rings may have the same zero-divisor graph and that the zero-divisor graph does not detect nilpotent elements.
436
ANDERSON AND LIVINGSTON
Žb. By part Ža. above, all connected graphs with less than four vertices may be realized as ⌫ Ž R .. Of the eleven graphs with four vertices, only six are connected. Of these six, only the following three graphs may be realized as ⌫ Ž R ..
We next sketch a proof that the graph ⌫ with vertices a, b, c, d4 and edges a y b, b y c, c y d cannot be realized as ⌫ Ž R .. Suppose that there is a ring R with ZŽ R . s 0, a, b, c, d4 and only the above zero-divisor relations. Then a q c g ZŽ R . since Ž a q c . b s 0. Hence a q c must be either 0, a, b, c, or d. A simple check yields a q c s b as the only possibility. Similarly, b q d s c. Hence b s a q c s a q b q d; so a q d s 0. Thus bd s bŽya. s 0, a contradiction. The proofs for the other two nonrealizable connected graphs on four vertices are similar. Žc. We have seen above that ⌫ Ž R . can be a triangle or a square. But, ⌫ Ž R . cannot be an n-gon for any n G 5. ŽThe proof is similar to that in part Žb. above; see w11, Proposition 17x for details. This also follows directly from Theorems 2.2 and 2.4.. However, for each n G 3, there is a zero-divisor graph with an n-cycle. Let R n s ⺪ 2 w X 1 , . . . , X n xrI s ⺪ 2 w x 1 , . . . , x n x, where I s Ž X 12 , . . . , X n2 , X 1 X 2 , X 2 X 3 , . . . , X n X 1 .. Then ⌫ Ž R n . is finite and has a cycle of length n, namely, x 1 y x 2 y ⭈⭈⭈ yx n y x 1. Žd. Let A and B be integral domains and let R s A = B. Then ⌫ Ž R . is a complete bipartite graph Ži.e., ⌫ Ž R . may be partitioned into two disjoint vertex sets V1 s Ž a, 0. < a g A*4 and V2 s Ž0, b . < b g B*4 and two vertices x and y are adjacent if and only if they are in distinct vertex sets. with < ⌫ Ž R .< s < A < q < B < y 2. The complete bipartite graph with vertex sets having m and n elements, respectively, will be denoted by K m, n. A complete bipartite graph of the form K 1, n is called a star graph. If A s ⺪ 2 , then ⌫ Ž R . is a star graph with < ⌫ Ž R .< s < B <. For example, ⌫ Ž⺖p = ⺖q . s K py 1, qy1 and ⌫ Ž⺪ 2 = ⺖q . s K 1, qy1. We give two specific examples.
ZERO-DIVISOR GRAPH
437
Of course, ⌫ Ž R . may be infinite Ži.e., a ring may have an infinite number of zero-divisors.. But ⌫ Ž R . is probably of most interest when it is finite, for then one can draw ⌫ Ž R .. We will state most results in as general a setting as possible, and then often specialize to the finite case. We next show that ⌫ Ž R . is finite Žexcept for the trivial case when ⌫ Ž R . is empty. only when R is itself finite. Thus we will often restrict to the case where R is a finite ring. Recall that if R is finite, then each element of R is either a unit or a zero-divisor, each prime ideal of R is an annihilator ideal, and each nonunit of R is nilpotent if and only if R is local. Moreover, if R is a finite local ring with maximal ideal M, then char R s p n for some prime p and integer n G 1. Hence M Žs ZŽ R .. is a p-group, so < ⌫ Ž R .< s p m y 1 for some integer m G 0. The essence of our first result is that ZŽ R . is finite if and only if either R is finite or an integral domain Žthis result, with a different proof, and the fact that < R < F < ZŽ R .< 2 when 2 F < ZŽ R .< - ⬁ are due to N. Ganesan w6, Theorem 1x; see w7x and w10x for noncommutative analogs.. THEOREM 2.2. Let R be a commutati¨ e ring. Then ⌫ Ž R . is finite if and only if either R is finite or an integral domain. In particular, if 1 F < ⌫ Ž R .< - ⬁, then R is finite and not a field. Proof. Suppose that ⌫ Ž R . Žs ZŽ R .*. is finite and nonempty. Then there are nonzero x, y g R with xy s 0. Let I s annŽ x .. Then I ; ZŽ R . is finite and ry g I for all r g R. If R is infinite, then there is an i g I with J s r g R < ry s i4 infinite. For any r, s g J, Ž r y s . y s 0, so annŽ y . ; ZŽ R . is infinite, a contradiction. Thus R must be finite. Recall that a graph ⌫ is connected if there is a path between any two distinct vertices. For distinct vertices x and y of ⌫, let dŽ x, y . be the length of the shortest path from x to y Ž dŽ x, y . s ⬁ if there is no such path.. The diameter of ⌫ is diamŽ ⌫ . s sup dŽ x, y .< x and y are distinct vertices of ⌫4 . The girth of ⌫, denoted by g Ž ⌫ ., is defined as the length of the shortest cycle in ⌫ Ž g Ž ⌫ . s ⬁ if ⌫ contains no cycles.. Note that if ⌫ contains a cycle, then g Ž ⌫ . F 2 diamŽ ⌫ . q 1 w5, Proposition 1.3.2x. We next show that the zero-divisor graphs are all connected and have exceed-
438
ANDERSON AND LIVINGSTON
ingly small ŽF 3. diameter and girth. Thus, for distinct x, y g ZŽ R .*, either xy s 0, xz s zy s 0 for some z g ZŽ R .* y x, y4 , or xz1 s z1 z 2 s z 2 y s 0 for some distinct z1 , z 2 g ZŽ R .* y x, y4 . THEOREM 2.3. Let R be a commutati¨ e ring. Then ⌫ Ž R . is connected and diamŽ ⌫ Ž R .. F 3. Moreo¨ er, if ⌫ Ž R . contains a cycle, then g Ž ⌫ Ž R .. F 7. Proof. Let x, y g ZŽ R .* be distinct. If xy s 0, then dŽ x, y . s 1. So suppose that xy is nonzero. If x 2 s y 2 s 0, then x y xy y y is a path of length 2; thus dŽ x, y . s 2. If x 2 s 0 and y 2 / 0, then there is a b g ZŽ R .* y x, y4 with by s 0. If bx s 0, then x y b y y is a path of length 2. If bx / 0, then x y bx y y is a path of length 2. In either case, dŽ x, y . s 2. A similar argument holds if y 2 s 0 and x 2 / 0. Thus we may assume that xy, x 2 , and y 2 are all nonzero. Hence there are a, b g ZŽ R .* y x, y4 with ax s by s 0. If a s b, then x y a y y is a path of length 2. Thus we may assume that a / b. If ab s 0, then x y a y b y y is a path of length 3, and hence dŽ x, y . F 3. If ab / 0, then x y ab y y is a path of length 2; thus dŽ x, y . s 2. Hence dŽ x, y . F 3, and thus diamŽ ⌫ Ž R .. F 3. The ‘‘moreover’’ statement follows from the comments preceding the theorem. Example 2.1Ža. gives several rings R with diamŽ ⌫ Ž R .. s 0, 1, or 2. In R s ⺪ 2 = ⺪ 4 , the path Ž0, 1. y Ž1, 0. y Ž0, 2. y Ž1, 2. shows that diamŽ ⌫ Ž R .. s 3. We next show that when R is Artinian, we can improve the ‘‘moreover’’ statement of Theorem 2.3 to g Ž ⌫ Ž R .. F 4. Example 2.1 gives several rings R with g Ž ⌫ Ž R .. s 3, 4, or ⬁. We know of no commutative ring R with 5 F g Ž ⌫ Ž R .. - ⬁, and we conjecture that none exist. THEOREM 2.4. Let R be a commutati¨ e Artinian ring Ž in particular, R could be a finite commutati¨ e ring .. If ⌫ Ž R . contains a cycle, then grŽ ⌫ Ž R .. F 4. Proof. Suppose that ⌫ Ž R . contains a cycle. By w2, Theorem 8.7x, R is a finite direct product of Artinian local rings. First, suppose that R is local with nonzero maximal ideal M. Then M s annŽ x . for some x g M* w9, Theorem 82x. If there are distinct y, z g M* y x 4 with yz s 0, then y y x y z y y is a triangle. Otherwise, ⌫ Ž R . contains no cycles, a contradiction. In this case, grŽ ⌫ Ž R .. s 3. Next, suppose that R s R1 = R 2 . If both < R1 < G 3 and < R 2 < G 3, then we may choose a i g R i y 0, 14 . Then Ž1, 0. y Ž0, 1. y Ž a1 , 0. y Ž0, a2 . y Ž1, 0. is a square. So in this case, grŽ ⌫ Ž R .. F 4. Thus we may assume that R1 s ⺪ 2 . If < ZŽ R 2 .< F 2, then ⌫ Ž R . contains no cycles, a contradiction. Hence we must have < ZŽ R 2 .< G 3. Since ⌫ Ž R . is connected, there are distinct x, y g ZŽ R 2 .* with xy s 0. Thus Ž0, x . y Ž1, 0. y Ž0, y . y Ž0, x . is a triangle. Hence in this case, grŽ ⌫ Ž R .. s 3. Thus in all cases, grŽ ⌫ Ž R .. F 4.
ZERO-DIVISOR GRAPH
439
The proof of the above theorem shows that a finite commutative ring R has grŽ ⌫ Ž R .. s 4 if and only if either R ( F = K for finite fields F and K with < F <, < K < G 3 Žcf. Example 2.1Žd.., or R ( F = A for F a finite field with < F < G 3 and A a finite ring with < ZŽ A.< s 2 Žso either A ( ⺪ 4 or ⺪ 2 w X xrŽ X 2 ... Also, one may easily show that grŽ ⌫ Ž R .. s ⬁ if and only if either < ⌫ Ž R .< F 2, < ⌫ Ž R .< s 3 and ⌫ Ž R . is not complete, or R ( ⺪ 2 = A for A a finite field or a finite ring with < ZŽ A.< s 2 Žcf. Theorem 2.13.. For each integer n G 1, let ⌫n be the graph with vertex set x 1 , . . . , x n4 and x 1 y x 2 , x 2 y x 3 , . . . , x ny1 y x n as its only edges. By parts Ža. and Žb. of Example 2.1 and Theorem 2.3, the ‘‘line graph’’ ⌫n can be realized as ⌫ Ž R . if and only if n F 3. We next determine when ⌫ Ž R . has a vertex adjacent to every other vertex Ži.e., when ⌫ Ž R . has a spanning tree which is a star graph.. Special cases of this are when either ⌫ Ž R . is a complete graph or a star graph. This is the key concept in characterizing these graphs. THEOREM 2.5. Let R be a commutati¨ e ring. Then there is a ¨ ertex of ⌫ Ž R . which is adjacent to e¨ ery other ¨ ertex if and only if either R ( ⺪ 2 = A, where A is an integral domain, or ZŽ R . is an annihilator ideal Ž and hence is prime.. Proof. Ž«. Suppose that ZŽ R . is not an annihilator ideal and 0 / a g ZŽ R . is adjacent to every other vertex. Now a f annŽ a. s I, for otherwise ZŽ R . s I would be an annihilator ideal. Thus I is maximal among annihilator ideals, and hence is prime w9, Theorem 6x. If a2 / a, then a3 s a2 a s 0, and hence a g I, a contradiction. Thus a2 s a; so R s Ra [ RŽ 1 y a.. Hence we may assume that R s R1 = R 2 with Ž1, 0. adjacent to every other vertex. For any 1 / c g R1 , Ž c, 0. is a zero-divisor, so Ž c, 0. s Ž c, 0.Ž1, 0. s Ž0, 0., a contradiction unless c s 0. Hence, R1 ( ⺪ 2 . If R 2 is not an integral domain, then there is a nonzero b g ZŽ R 2 .. Then Ž1, b . is a zero-divisor of R which is not adjacent to Ž1, 0., a contradiction. Thus R 2 must be an integral domain. Note that if ZŽ R . is an annihilator ideal, then it is certainly maximal among annihilator ideals, and hence is prime. Ž¥. If R ( ⺪ 2 = A for A an integral domain, then Ž1, 0. is adjacent to every other vertex. If ZŽ R . s annŽ x . for some nonzero x g R, then x is adjacent to every other vertex. Example 2.1Ža. shows that both cases in Theorem 2.5 may occur for the same graph. However, if R is reduced and ⌫ Ž R . has a vertex adjacent to every other vertex, then R must have the form ⺪ 2 = A for A an integral domain. Note that the proof of Theorem 2.5 shows that if a vertex x of ⌫ Ž R . is adjacent to every other vertex, then either x is idempotent with Rx s 0, x 4
440
ANDERSON AND LIVINGSTON
a prime ideal of R, or ZŽ R . s annŽ x .. In fact, if ZŽ R . is an annihilator ideal, then annŽ ZŽ R ..* is precisely the set of vertices which are adjacent to every other vertex. Suppose that R is quasilocal with Žnonzero. maximal ideal M. If there is a least positive integer k with M k s 0, then ZŽ R . s M s annŽ x . for any nonzero x g M ky 1 Žso if < RrM < G 3, then there at least two vertices adjacent to every other vertex.. However, easy examples show that we need not have annŽ M . s M ky 1. Since ZŽ R . is always a union of prime ideals w9, p. 3x, ZŽ R . is a prime ideal if Žand only if. it is an ideal. If R is also Noetherian, then ZŽ R . is an annihilator ideal if and only if it is an Žprime. ideal w9, Theorems 6 and 82x. Recall that 04 is a primary ideal of R if and only if ZŽ R . s nilŽ R .. If dim R s 0, then ZŽ R . s nilŽ R . if and only if ZŽ R . is a Žprime. ideal of R; if R is finite, this is equivalent to R being local. COROLLARY 2.6. Let R be a commutati¨ e Noetherian ring. Then there is a ¨ ertex of ⌫ Ž R . which is adjacent to e¨ ery other ¨ ertex if and only if either
R ( ⺪ 2 = A, where A is an Ž Noetherian. integral domain, or ZŽ R . is an Ž prime. ideal of R. If in addition, dim R s 0, then either R ( ⺪ 2 = A, where A is a field, or 04 is a primary ideal of R Ž i.e., ZŽ R . s nilŽ R ... The Noetherian hypothesis is needed in Corollary 2.6. Let X 4 be an infinite family of indeterminates. Then R s ⺪ 2 w X 4xrŽ X2 4. is a nonNoetherian zero-dimensional quasilocal ring with maximal ideal ZŽ R . s nilŽ R . s Ž X 4.rŽ X2 4., but ZŽ R . is not an annihilator ideal. We next specialize to the case when R is finite.
COROLLARY 2.7. Let R be a finite commutati¨ e ring. Then there is a ¨ ertex of ⌫ Ž R . which is adjacent to e¨ ery other ¨ ertex if and only if either R ( ⺪ 2 = F, where F is a finite field, or R is local. Moreo¨ er, for some prime p and integer n G 1, < ⌫ Ž R .< s < F < s p n if R ( ⺪ 2 = F, and < ⌫ Ž R .< s p n y 1 if R is local.
We next determine when ⌫ Ž R . is a complete graph Ži.e., any two vertices are adjacent.. By definition, ⌫ Ž R . is complete if and only if xy s 0 for all distinct x, y g ZŽ R .. Except for the case when R ( ⺪ 2 = ⺪ 2 , our next theorem shows that we must also have x 2 s 0 for all x g ZŽ R . when ⌫ Ž R . is complete. So, except for that one case, nilpotent elements are detected by complete graphs. We denote the complete graph on n vertices by K n. THEOREM 2.8. Let R be a commutati¨ e ring. Then ⌫ Ž R . is complete if and only if either R ( ⺪ 2 = ⺪ 2 or xy s 0 for all x, y g ZŽ R .. Proof. Ž¥. By definition. Ž«. Suppose that ⌫ Ž R . is complete, but there is an x g ZŽ R . with x 2 / 0. We show that x 2 s x. If not, then x 3 s x 2 x s 0. Hence x 2 Ž x q x 2 . s 0 with x 2 / 0, so x q x 2 g ZŽ R .. If x q x 2 s x, then x 2 s 0, a contra-
ZERO-DIVISOR GRAPH
441
diction. Thus x q x 2 / x, so x 2 s x 2 q x 3 s x Ž x q x 2 . s 0 since ⌫ Ž R . is complete, again a contradiction. Hence x 2 s x. As in the proof of Theorem 2.5, we have R ( ⺪ 2 = A, and necessarily A ( ⺪ 2 . COROLLARY 2.9. Let R be a commutati¨ e ring. For x, y g ZŽ R ., define x ; y if xy s 0 or x s y, and define x ;* y if xy s 0. Ža. The relation ; is transiti¨ e Ž equi¨ alently, an equi¨ alence relation. if and only if ⌫ Ž R . is complete. Žb. The relation ;* is transiti¨ e Ž equi¨ alently, an equi¨ alence relation. if and only if ⌫ Ž R . is complete and R \ ⺪ 2 = ⺪ 2 . Proof. Both parts follow directly from Theorems 2.3 and 2.8. Suppose that R is a finite commutative ring and not isomorphic to ⺪ 2 = ⺪ 2 . By Corollary 2.7 and Theorem 2.8, ⌫ Ž R . is complete if and only if R is local with maximal ideal M and M 2 s 0. However, this does not hold if R is not finite. For example, if ZŽ R . 2 s 0, then by McCoy’s Theorem, also ZŽ Rw X x. 2 s 0. Hence ⌫ Ž⺪ 4 w X x. Žs 2⺪ 4 w X x y 04. is an infinite complete graph, but ⺪ 4 w X x is not local. Suppose that ⌫ Ž R . is complete and R \ ⺪ 2 = ⺪ 2 ; then by Theorem 2.8, ZŽ R . is an ideal of R with ZŽ R . 2 s 0. Hence ZŽ R . s nilŽ R . is a prime ideal of R. THEOREM 2.10. Let R be a finite commutati¨ e ring. If ⌫ Ž R . is complete, then either R ( ⺪ 2 = ⺪ 2 or R is local with char R s p or p 2 , and < ⌫ Ž R .< s p n y 1, where p is prime and n G 1. Proof. For a field F, the graph ⌫ Ž⺪ 2 = F . is not complete unless F s ⺪ 2 . So assume that R \ ⺪ 2 = ⺪ 2 . By Corollary 2.7, R must be local with maximal ideal M. Hence char R s p m for some prime p and integer m G 1. If m G 3, then R would have nonadjacent zero-divisors, a contradiction. Hence char R s p or p 2 . Since M is a p-group, < ⌫ Ž R .< s p n y 1 for some prime p and integer n G 1. We next show conversely that for each prime p and integer n G 1, there is a ring R such that ⌫ Ž R . is complete and < ⌫ Ž R .< s p n y 1. Thus K m may be realized as ⌫ Ž R . if and only if m s p n y 1 for some prime p and integer n G 1. Example 2.11 also shows that any graph ⌫ is a subgraph of a zero-divisor graph ⌫ Ž R . for some Žquasilocal. commutative ring R. EXAMPLE 2.11. Ža. Let p be prime, X s X␣ 4␣ g ᑜ a family of indeterminates, A s ⺪ p w X x, and I s Ž X␣ X < ␣ ,  g ᑜ 4.. Then R s ArI is a zero-dimensional quasilocal ring with maximal ideal M s ŽX.rI and M 2 s 0. Thus ⌫ Ž R . is complete. Moreover, < ⌫ Ž R .< s p n y 1 if < ᑜ < s n and < ⌫ Ž R .< s < ᑜ < if ᑜ is infinite. Also, for any field F, the zero-dimensional
442
ANDERSON AND LIVINGSTON
local ring S s F w X xrŽ X 2 . has a complete zero-divisor graph with < ⌫ Ž S .< s < F < y 1. Žb. Let p be prime and R s ⺪ p 2 . Then ⌫ Ž R . is complete with < ⌫ Ž R .< s p y 1. Note that ⺪ p 2 and ⺪ p w X xrŽ X 2 . are not isomorphic as rings, but have the same zero-divisor graph, namely, K py 1. In particular, none of ⺪ 2 = ⺪ 2 , ⺪ 9 , and ⺪ 3w X xrŽ X 2 . are isomorphic as rings, but all have the same complete zero-divisor graph K 2 . We next consider when ⌫ Ž R . has exactly one vertex which is adjacent to every other vertex. LEMMA 2.12. Let R be a finite commutati¨ e ring. If ⌫ Ž R . has exactly one ¨ ertex adjacent to e¨ ery other ¨ ertex and no other adjacent ¨ ertices, then either
R ( ⺪ 2 = F, where F is a finite field with < F < G 3, or R is local with maximal ideal M satisfying RrM ( ⺪ 2 , M 3 s 0, and < M 2 < F 2. Thus < ⌫ Ž R .< is either p n or 2 n y 1 for some prime p and integer n G 1. Proof. If R \ ⺪ 2 = F, then R is local with maximal ideal M by Corollary 2.7. Thus M s annŽ x . for a unique x g M. Let k be the least positive integer with M k s 0. Then M s annŽ y . for any nonzero y g M ky 1. Hence M ky1 s 0, x 4 , and thus < M ky 1rM k < s 2 yields RrM ( ⺪ 2 . If k G 4, then < M ky 2 < G 4. Hence for any distinct a, b g M ky 2 y M ky1 , ab g M 2 ky4 ; M k . Thus ab s 0, a contradiction. Hence M 3 s 0, and thus < M 2 < F 2. ŽNote that we may have M 2 s 0.. In Example 2.1Žd., we observed that ⌫ Ž⺪ 2 = F . is a star graph of order < F < when F is a finite field. We next use the previous lemma to show that, except for zero-divisor graphs of small order Žsee Example 2.1Ža.., this is the only way a star graph arises as ⌫ Ž R .. THEOREM 2.13. Let R be a finite commutati¨ e ring with < ⌫ Ž R .< G 4. Then ⌫ Ž R . is a star graph if and only if R ( ⺪ 2 = F, where F is a finite field. In particular, if ⌫ Ž R . is a star graph, then < ⌫ Ž R .< s p n for some prime p and integer n G 0. Con¨ ersely, each star graph of order p n can be realized as ⌫ Ž R ..
Proof. Suppose that ⌫ Ž R . is a star graph and R \ ⺪ 2 = F. By Corollary 2.7 and Lemma 2.12, we may assume that Ž R, M . is local with < M < s 2 k for some k G 3 Žsince M is a 2-group and < ⌫ Ž R .< G 4. and < M 2 < s 2. Let M s annŽ x . and choose distinct a, b, c, d g M* y x 4 . Then ab s ac s ad s x since M 2 s 0, x 4 and there are no other zero-divisor relations. Hence aŽ b y c . s aŽ b y d . s 0. Note that annŽ a. s 0, x 4 Ž a f annŽ a. since annŽ a. has even order. and thus b y c s b y d s x. Hence c s d, a contradiction. The ‘‘in particular’’ statement is clear and the ‘‘conversely’’ statement follows from the remarks preceding the theorem.
ZERO-DIVISOR GRAPH
443
EXAMPLE 2.14. For each integer n G 1, let R n s ⺪ 2 w X xrŽ X nq 1 . s ⺪ 2 w x x, a finite local ring. Then x n is the only vertex adjacent to every other vertex. However, for n G 3, ⌫ Ž R n . is not a star graph since the vertices x ny 1 q x n and x ny1 are also adjacent. Note that < ⌫ Ž R n .< s 2 n y 1.
3. AUTOMORPHISMS OF ⌫ Ž R . In this section, we study AutŽ ⌫ Ž R .., the group of graph automorphisms of ⌫ Ž R .. We first show that distinct ring automorphisms of R induce distinct graph automorphisms of ⌫ Ž R . when R is a finite ring which is not a field, and then we determine AutŽ ⌫ Ž R .. for several specific classes of finite rings. In particular, we compute AutŽ ⌫ Ž⺪ n ... Recall that a graph automorphism f of a graph ⌫ is a bijection f : ⌫ ª ⌫ which preserves adjacency. The set AutŽ ⌫ . of all graph automorphisms of ⌫ forms a group under the usual composition of functions. If < ⌫ < s n, then in the obvious way AutŽ ⌫ . is isomorphic to a subgroup of Sn , and clearly AutŽ K n . ( Sn . In fact, for a graph ⌫ of order n, AutŽ ⌫ . ( Sn if and only if ⌫ s K n. In the obvious way, each ring automorphism of R induces a graph automorphism of ⌫ Ž R .; we thus have a natural group homomorphism : AutŽ R . ª AutŽ ⌫ Ž R .. given by just restricting each f g AutŽ R . to ZŽ R .*. In general, AutŽ ⌫ Ž R .. is much larger than AutŽ R .; for example, each AutŽ⺪ n . is trivial. We first show that if R is a finite ring which is not a field, then : AutŽ R . ª AutŽ ⌫ Ž R .. is injective, i.e., an f g AutŽ R . is completely determined by its action on ZŽ R .. THEOREM 3.1. Let R be a finite commutati¨ e ring which is not a field and let f g AutŽ R .. If f Ž x . s x for all x g ZŽ R ., then f s 1 R . Thus : AutŽ R . ª AutŽ ⌫ Ž R .. is a monomorphism.
444
ANDERSON AND LIVINGSTON
Proof. First suppose that R has distinct maximal ideals M and N. Then M, N ; ZŽ R .. Since R s M q N, we must have f s 1 R . Thus we may assume that R is local with nonzero maximal ideal M Žs ZŽ R ... Let char R s p n and < RrM < s p m for some prime p and integers n, m G 1. We show that f Ž u. s u for each u g UŽ R .. Observe that for any m g M, um s f Ž um. s f Ž u. m, and hence Ž f Ž u. y u. m s 0. Thus f Ž u. s u q x for some x g M Žnote that x may depend on u. with xM s 0. Since < RrM < s m m p m , we have u p y1 s 1 q y for some y g M, and hence f fixes u p y1 . m m m m p my1 p y1 p y1 p y1 p y1 m . s f Ž u. Thus u s fŽu s Žu q x. su qŽp y m m 1.Ž u p y2 . x since xM s 0, and hence Ž p m y 1.Ž u p y2 . x s 0. Since p m y 1 m and u p y2 are units in R, we have x s 0. Thus f Ž u. s u for all u g UŽ R ., and hence f s 1 R . We next show that AutŽ ⌫ Ž⺪ n .. is a Žfinite. direct product of symmetric groups, and thus ⌫ Ž⺪ n . is highly symmetrical. The proof of this result gives the explicit construction of AutŽ ⌫ Ž⺪ n .; specific examples are given in parts Žc. and Žd. of Example 3.4. For any graph ⌫, the degree of a vertex x of ⌫ is ␦ Ž x . s < y g ⌫ < y is adjacent to x 4<. For a vertex x of a zero-divisor graph ⌫ Ž R ., we have ␦ Ž x . s
ZERO-DIVISOR GRAPH
445
Ž f < Vd ., with f < Vd viewed in the natural way as an element of Sn . By the d above comments, is a well-defined group monomorphism. To show that is surjective, it suffices to show that for each fixed d g X and permutation ␣ of Vd , there is an f g AutŽ ⌫ Ž⺪ n .. with f < Vd s ␣ and f < Vd⬘ s 1Vd⬘ for all d⬘ / d in X. This follows since for any x, y g Vd and a g ⺪ n , ax s 0 if and only if ay s 0. It is well known that any finite group G may be realized as AutŽ ⌫ . for some graph ⌫. It would be interesting to know exactly which groups may be realized as AutŽ ⌫ Ž R .. or AutŽ ⌫ Ž⺪ n ... The next corollary gives a partial answer for AutŽ ⌫ Ž⺪ n ... COROLLARY 3.3.
Let n G 4 be a nonprime integer. Then
Ža. AutŽ ⌫ Ž⺪ n .. is tri¨ ial if and only if n s 4. Žb. AutŽ ⌫ Ž⺪ n .. is abelian if and only if n s 4, 6, 8, 9, or 12. In particular, Aut Ž ⌫ Ž⺪ n .. ( ⺪ 2 when n s 6, 8, or 9, and AutŽ ⌫ Ž⺪12 .. ( ⺪ 2 = ⺪2 = ⺪2. Proof. The computations of AutŽ ⌫ Ž⺪ n .. for n s 4, 6, 8, and 9 follow easily from Theorem 3.2 or Example 2.1; for n s 12, see Example 3.4Žc.. We show that AutŽ ⌫ Ž⺪ n .. is nonabelian for all other nonprime integers n G 4. It is sufficient to show that in each case, some < Vd < G 3, and hence AutŽ ⌫ Ž⺪ n .. has a direct factor of S m for some m G 3. If n s 2 r with r G 4, then 2, 6, 104 ; V2 . If n s 3 r with r G 3, then 3, 6, 154 ; V3 . If n s p r with p G 5 prime, then p, 2 p, 3 p4 ; Vp . Let n s p1r 1 ⭈⭈⭈ pkr k be its prime factorization. If p1 G 3 and k G 2, then p1 , 2 p1, 4 p14 ; Vp . Thus we may assume that p1 s 2 with r s r 1 G 1 and 1 k G 2. If n ) 2 r ⭈ 3, then 2 r , 2 rq1 , 2 rq2 4 ; V2 r . Hence we may assume that n s 2 r ⭈ 3. If r G 3, then 2, 10, 144 ; V2 . Thus AutŽ ⌫ Ž⺪ n .. is abelian only if n s 4, 8, 9, 6, or 12. We conclude the paper with some specific examples of AutŽ ⌫ Ž R ... EXAMPLE 3.4. Ža. In Example 2.1Ža., we saw that the zero-divisor graphs of ⺖4 w X xrŽ X 2 . and ⺪ 3 = ⺪ 3 are a triangle and a square, respectively. Thus AutŽ ⌫ Ž⺖4 w X xrŽ X 2 ... ( S3 and AutŽ ⌫ Ž⺪ 3 = ⺪ 3 .. ( A 4 Ž( S2 wrS2 , where wr denotes the wreath product.. Žb. Let F and K be finite fields with distinct orders m and n, respectively. As in Example 2.1Žd., ⌫ Ž F = K . is the complete bipartite graph K my 1, ny1. Hence AutŽ ⌫ Ž F = K .. ( Smy 1 = Sny1 has order Ž m y 1.!Ž n y 1.!. Also, ⌫ Ž F = F . is the complete bipartite graph K my 1, my1. It may be shown that AutŽ K my 1, my1 . ( Smy 1wrS2 , which has order 2ŽŽ m y 1.!. 2 .
446
ANDERSON AND LIVINGSTON
Žc. We illustrate the proof of Theorem 3.2 by computing AutŽ ⌫ Ž⺪ 12 ... We have X s 2, 3, 4, 64 , and thus V2 s 2, 104 , V3 s 3, 94 , V4 s 4, 84 , and V6 s 64 . Hence AutŽ ⌫ Ž⺪12 .. ( S2 = S2 = S2 = S1 ( ⺪ 2 = ⺪ 2 = ⺪ 2 . This is also evident from the obvious symmetries of the zero-divisor graph of ⺪ 12 below.
Žd. For a more complicated example, let R s ⺪1225 Ž( ⺪ 25 = ⺪ 49 . with < ⌫ Ž R .< s 384. Then X s 5, 7, 25, 35, 49, 175, 2454 , and one may calculate that < V5 < s 168, < V7 < s 120, < V25 < s 42, < V35 < s 24, < V49 < s 20, < V175 < s 6, and < V245 < s 4. Thus AutŽ ⌫ Ž R .. ( S168 = S120 = S42 = S24 = S20 = S6 = S4 has order 168!⭈ 120!⭈ 42!⭈ 24!⭈ 6!⭈ 4!, which is approximately 6.192877 = 10 598. Note added in proof. In the paper, ‘‘Cycles of zero-divisors,’’ S. B. Mulay has shown, among other things, that Theorem 2.4 does indeed hold for any commutative ring.
ACKNOWLEDGMENTS Part of this paper is from w11x, which contains many more examples. We would like to thank Richard M. Stafford of the NSA, who also helped with calculations using Magma, and Aaron Lauve, who participated in an NSF-sponsored REU program with the first author at the University of Tennessee during the summer of 1998, for their many helpful discussions.
REFERENCES 1. D. D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra 159 Ž1993., 500᎐514.
ZERO-DIVISOR GRAPH
447
2. M. F. Atiyah and I. G. MacDonald, ‘‘Introduction to Commutative Algebra,’’ Addison᎐Wesley, Reading, MA, 1969. 3. I. Beck, Coloring of commutative rings, J. Algebra 116 Ž1988., 208᎐226. 4. B. Bollabas, ´ ‘‘Graph Theory, An Introductory Course,’’ Springer-Verlag, New York, 1979. 5. R. Diestel, ‘‘Graph Theory,’’ Springer-Verlag, New York, 1997. 6. N. Ganesan, Properties of rings with a finite number of zero-divisors, Math. Ann. 157 Ž1964., 215᎐218. 7. N. Ganesan, Properties of rings with a finite number of zero-divisors, II, Math. Ann. 161 Ž1965., 241᎐246. 8. F. Harary, ‘‘Graph Theory,’’ Addison᎐Wesley, Reading, MA, 1972. 9. I. Kaplansky, ‘‘Commutative Rings,’’ rev. ed., Univ. of Chicago Press, Chicago, 1974. 10. K. Koh, On ‘‘Properties of rings with a finite number of zero-divisors,’’ Math. Ann. 171 Ž1967., 79᎐80. 11. P. S. Livingston, ‘‘Structure in Zero-divisor Graphs of Commutative Rings,’’ Masters Thesis, The University of Tennessee, Knoxville, TN, December 1997. 12. B. R. McDonald, ‘‘Finite Rings with Identity,’’ Dekker, New York, 1974.
The Zero-Divisor Graph of a Commutative Ring David F. Anderson and Philip S. Livingston Mathematics Department, The Uni¨ ersity of Tennessee, Knox¨ ille, Tennessee 37996 E-mail: [email protected], [email protected] Communicated by Kent R. Fuller Received October 5, 1998
For each commutative ring R we associate a Žsimple. graph ⌫ Ž R .. We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of ⌫ Ž R .. 䊚 1999 Academic Press
1. INTRODUCTION Let R be a commutative ring Žwith 1. and let ZŽ R . be its set of zero-divisors. We associate a Žsimple. graph ⌫ Ž R . to R with vertices ZŽ R .* s ZŽ R . y 04 , the set of nonzero zero-divisors of R, and for distinct x, y g ZŽ R .*, the vertices x and y are adjacent if and only if xy s 0. Thus ⌫ Ž R . is the empty graph if and only if R is an integral domain. The main object of this paper is to study the interplay of ring-theoretic properties of R with graph-theoretic properties of ⌫ Ž R .. This study helps illuminate the structure of ZŽ R .. For x, y g ZŽ R ., define x ; y if xy s 0 or x s y. The relation ; is always reflexive and symmetric, but is usually not transitive. The zero-divisor graph ⌫ Ž R . measures this lack of transitivity in the sense that ; is transitive if and only if ⌫ Ž R . is complete. The idea of a zero-divisor graph of a commutative ring was introduced by I. Beck in w2x, where he was mainly interested in colorings. This investigation of colorings of a commutative ring was then continued by D. D. Anderson and M. Naseer in w1x. Their definition was slightly different than ours; they let all elements of R be vertices and had distinct x and y adjacent if and only if xy s 0. We will denote their zero-divisor graph of R by ⌫0 Ž R .. In ⌫0 Ž R ., the vertex 0 is adjacent to every other vertex, but non-zero-divisors are adjacent only to 0. Note that ⌫ Ž R . is a Žinduced. subgraph of ⌫0 Ž R .. Our results for ⌫ Ž R . have natural analogs to ⌫0 Ž R .; 434 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
ZERO-DIVISOR GRAPH
435
however, we feel that our definition better illustrates the zero-divisor structure of R. In Section 2, we give many examples, we show that ⌫ Ž R . is always connected with diamŽ ⌫ Ž R .. F 3, and we determine when ⌫ Ž R . is a complete graph or a star graph. A key step is characterizing when a vertex is adjacent to every other vertex. In the third section, we study the automorphism group of ⌫ Ž R .. We will include basic definitions from graph theory as needed. Basic references for graph theory are w4, 5, 8x; for commutative ring theory, see w2, 9, 12x. All rings R are commutative with 1 Ž/ 0., and for A ; R, we let A* s A y 04 . As usual, the rings of integers and integers modulo n will be denoted by ⺪ and ⺪ n , respectively, and ⺖q will be the finite field with q elements. To avoid trivialities when ⌫ Ž R . is empty, we will implicitly assume when necessary that R is not an integral domain. 2. PROPERTIES OF ⌫ Ž R . In this section, we show that ⌫ Ž R . is always connected and has small diameter and girth, and we determine which complete graphs and star graphs may be realized as ⌫ Ž R .. We start with some examples which motivate later results Žalso see Examples 2.11 and 2.14.. Easy details Žincluding labeling the vertices. are left to the reader. EXAMPLE 2.1. Ža. Below are the zero-divisor graphs for several rings. Note that these examples show that nonisomorphic rings may have the same zero-divisor graph and that the zero-divisor graph does not detect nilpotent elements.
436
ANDERSON AND LIVINGSTON
Žb. By part Ža. above, all connected graphs with less than four vertices may be realized as ⌫ Ž R .. Of the eleven graphs with four vertices, only six are connected. Of these six, only the following three graphs may be realized as ⌫ Ž R ..
We next sketch a proof that the graph ⌫ with vertices a, b, c, d4 and edges a y b, b y c, c y d cannot be realized as ⌫ Ž R .. Suppose that there is a ring R with ZŽ R . s 0, a, b, c, d4 and only the above zero-divisor relations. Then a q c g ZŽ R . since Ž a q c . b s 0. Hence a q c must be either 0, a, b, c, or d. A simple check yields a q c s b as the only possibility. Similarly, b q d s c. Hence b s a q c s a q b q d; so a q d s 0. Thus bd s bŽya. s 0, a contradiction. The proofs for the other two nonrealizable connected graphs on four vertices are similar. Žc. We have seen above that ⌫ Ž R . can be a triangle or a square. But, ⌫ Ž R . cannot be an n-gon for any n G 5. ŽThe proof is similar to that in part Žb. above; see w11, Proposition 17x for details. This also follows directly from Theorems 2.2 and 2.4.. However, for each n G 3, there is a zero-divisor graph with an n-cycle. Let R n s ⺪ 2 w X 1 , . . . , X n xrI s ⺪ 2 w x 1 , . . . , x n x, where I s Ž X 12 , . . . , X n2 , X 1 X 2 , X 2 X 3 , . . . , X n X 1 .. Then ⌫ Ž R n . is finite and has a cycle of length n, namely, x 1 y x 2 y ⭈⭈⭈ yx n y x 1. Žd. Let A and B be integral domains and let R s A = B. Then ⌫ Ž R . is a complete bipartite graph Ži.e., ⌫ Ž R . may be partitioned into two disjoint vertex sets V1 s Ž a, 0. < a g A*4 and V2 s Ž0, b . < b g B*4 and two vertices x and y are adjacent if and only if they are in distinct vertex sets. with < ⌫ Ž R .< s < A < q < B < y 2. The complete bipartite graph with vertex sets having m and n elements, respectively, will be denoted by K m, n. A complete bipartite graph of the form K 1, n is called a star graph. If A s ⺪ 2 , then ⌫ Ž R . is a star graph with < ⌫ Ž R .< s < B <. For example, ⌫ Ž⺖p = ⺖q . s K py 1, qy1 and ⌫ Ž⺪ 2 = ⺖q . s K 1, qy1. We give two specific examples.
ZERO-DIVISOR GRAPH
437
Of course, ⌫ Ž R . may be infinite Ži.e., a ring may have an infinite number of zero-divisors.. But ⌫ Ž R . is probably of most interest when it is finite, for then one can draw ⌫ Ž R .. We will state most results in as general a setting as possible, and then often specialize to the finite case. We next show that ⌫ Ž R . is finite Žexcept for the trivial case when ⌫ Ž R . is empty. only when R is itself finite. Thus we will often restrict to the case where R is a finite ring. Recall that if R is finite, then each element of R is either a unit or a zero-divisor, each prime ideal of R is an annihilator ideal, and each nonunit of R is nilpotent if and only if R is local. Moreover, if R is a finite local ring with maximal ideal M, then char R s p n for some prime p and integer n G 1. Hence M Žs ZŽ R .. is a p-group, so < ⌫ Ž R .< s p m y 1 for some integer m G 0. The essence of our first result is that ZŽ R . is finite if and only if either R is finite or an integral domain Žthis result, with a different proof, and the fact that < R < F < ZŽ R .< 2 when 2 F < ZŽ R .< - ⬁ are due to N. Ganesan w6, Theorem 1x; see w7x and w10x for noncommutative analogs.. THEOREM 2.2. Let R be a commutati¨ e ring. Then ⌫ Ž R . is finite if and only if either R is finite or an integral domain. In particular, if 1 F < ⌫ Ž R .< - ⬁, then R is finite and not a field. Proof. Suppose that ⌫ Ž R . Žs ZŽ R .*. is finite and nonempty. Then there are nonzero x, y g R with xy s 0. Let I s annŽ x .. Then I ; ZŽ R . is finite and ry g I for all r g R. If R is infinite, then there is an i g I with J s r g R < ry s i4 infinite. For any r, s g J, Ž r y s . y s 0, so annŽ y . ; ZŽ R . is infinite, a contradiction. Thus R must be finite. Recall that a graph ⌫ is connected if there is a path between any two distinct vertices. For distinct vertices x and y of ⌫, let dŽ x, y . be the length of the shortest path from x to y Ž dŽ x, y . s ⬁ if there is no such path.. The diameter of ⌫ is diamŽ ⌫ . s sup dŽ x, y .< x and y are distinct vertices of ⌫4 . The girth of ⌫, denoted by g Ž ⌫ ., is defined as the length of the shortest cycle in ⌫ Ž g Ž ⌫ . s ⬁ if ⌫ contains no cycles.. Note that if ⌫ contains a cycle, then g Ž ⌫ . F 2 diamŽ ⌫ . q 1 w5, Proposition 1.3.2x. We next show that the zero-divisor graphs are all connected and have exceed-
438
ANDERSON AND LIVINGSTON
ingly small ŽF 3. diameter and girth. Thus, for distinct x, y g ZŽ R .*, either xy s 0, xz s zy s 0 for some z g ZŽ R .* y x, y4 , or xz1 s z1 z 2 s z 2 y s 0 for some distinct z1 , z 2 g ZŽ R .* y x, y4 . THEOREM 2.3. Let R be a commutati¨ e ring. Then ⌫ Ž R . is connected and diamŽ ⌫ Ž R .. F 3. Moreo¨ er, if ⌫ Ž R . contains a cycle, then g Ž ⌫ Ž R .. F 7. Proof. Let x, y g ZŽ R .* be distinct. If xy s 0, then dŽ x, y . s 1. So suppose that xy is nonzero. If x 2 s y 2 s 0, then x y xy y y is a path of length 2; thus dŽ x, y . s 2. If x 2 s 0 and y 2 / 0, then there is a b g ZŽ R .* y x, y4 with by s 0. If bx s 0, then x y b y y is a path of length 2. If bx / 0, then x y bx y y is a path of length 2. In either case, dŽ x, y . s 2. A similar argument holds if y 2 s 0 and x 2 / 0. Thus we may assume that xy, x 2 , and y 2 are all nonzero. Hence there are a, b g ZŽ R .* y x, y4 with ax s by s 0. If a s b, then x y a y y is a path of length 2. Thus we may assume that a / b. If ab s 0, then x y a y b y y is a path of length 3, and hence dŽ x, y . F 3. If ab / 0, then x y ab y y is a path of length 2; thus dŽ x, y . s 2. Hence dŽ x, y . F 3, and thus diamŽ ⌫ Ž R .. F 3. The ‘‘moreover’’ statement follows from the comments preceding the theorem. Example 2.1Ža. gives several rings R with diamŽ ⌫ Ž R .. s 0, 1, or 2. In R s ⺪ 2 = ⺪ 4 , the path Ž0, 1. y Ž1, 0. y Ž0, 2. y Ž1, 2. shows that diamŽ ⌫ Ž R .. s 3. We next show that when R is Artinian, we can improve the ‘‘moreover’’ statement of Theorem 2.3 to g Ž ⌫ Ž R .. F 4. Example 2.1 gives several rings R with g Ž ⌫ Ž R .. s 3, 4, or ⬁. We know of no commutative ring R with 5 F g Ž ⌫ Ž R .. - ⬁, and we conjecture that none exist. THEOREM 2.4. Let R be a commutati¨ e Artinian ring Ž in particular, R could be a finite commutati¨ e ring .. If ⌫ Ž R . contains a cycle, then grŽ ⌫ Ž R .. F 4. Proof. Suppose that ⌫ Ž R . contains a cycle. By w2, Theorem 8.7x, R is a finite direct product of Artinian local rings. First, suppose that R is local with nonzero maximal ideal M. Then M s annŽ x . for some x g M* w9, Theorem 82x. If there are distinct y, z g M* y x 4 with yz s 0, then y y x y z y y is a triangle. Otherwise, ⌫ Ž R . contains no cycles, a contradiction. In this case, grŽ ⌫ Ž R .. s 3. Next, suppose that R s R1 = R 2 . If both < R1 < G 3 and < R 2 < G 3, then we may choose a i g R i y 0, 14 . Then Ž1, 0. y Ž0, 1. y Ž a1 , 0. y Ž0, a2 . y Ž1, 0. is a square. So in this case, grŽ ⌫ Ž R .. F 4. Thus we may assume that R1 s ⺪ 2 . If < ZŽ R 2 .< F 2, then ⌫ Ž R . contains no cycles, a contradiction. Hence we must have < ZŽ R 2 .< G 3. Since ⌫ Ž R . is connected, there are distinct x, y g ZŽ R 2 .* with xy s 0. Thus Ž0, x . y Ž1, 0. y Ž0, y . y Ž0, x . is a triangle. Hence in this case, grŽ ⌫ Ž R .. s 3. Thus in all cases, grŽ ⌫ Ž R .. F 4.
ZERO-DIVISOR GRAPH
439
The proof of the above theorem shows that a finite commutative ring R has grŽ ⌫ Ž R .. s 4 if and only if either R ( F = K for finite fields F and K with < F <, < K < G 3 Žcf. Example 2.1Žd.., or R ( F = A for F a finite field with < F < G 3 and A a finite ring with < ZŽ A.< s 2 Žso either A ( ⺪ 4 or ⺪ 2 w X xrŽ X 2 ... Also, one may easily show that grŽ ⌫ Ž R .. s ⬁ if and only if either < ⌫ Ž R .< F 2, < ⌫ Ž R .< s 3 and ⌫ Ž R . is not complete, or R ( ⺪ 2 = A for A a finite field or a finite ring with < ZŽ A.< s 2 Žcf. Theorem 2.13.. For each integer n G 1, let ⌫n be the graph with vertex set x 1 , . . . , x n4 and x 1 y x 2 , x 2 y x 3 , . . . , x ny1 y x n as its only edges. By parts Ža. and Žb. of Example 2.1 and Theorem 2.3, the ‘‘line graph’’ ⌫n can be realized as ⌫ Ž R . if and only if n F 3. We next determine when ⌫ Ž R . has a vertex adjacent to every other vertex Ži.e., when ⌫ Ž R . has a spanning tree which is a star graph.. Special cases of this are when either ⌫ Ž R . is a complete graph or a star graph. This is the key concept in characterizing these graphs. THEOREM 2.5. Let R be a commutati¨ e ring. Then there is a ¨ ertex of ⌫ Ž R . which is adjacent to e¨ ery other ¨ ertex if and only if either R ( ⺪ 2 = A, where A is an integral domain, or ZŽ R . is an annihilator ideal Ž and hence is prime.. Proof. Ž«. Suppose that ZŽ R . is not an annihilator ideal and 0 / a g ZŽ R . is adjacent to every other vertex. Now a f annŽ a. s I, for otherwise ZŽ R . s I would be an annihilator ideal. Thus I is maximal among annihilator ideals, and hence is prime w9, Theorem 6x. If a2 / a, then a3 s a2 a s 0, and hence a g I, a contradiction. Thus a2 s a; so R s Ra [ RŽ 1 y a.. Hence we may assume that R s R1 = R 2 with Ž1, 0. adjacent to every other vertex. For any 1 / c g R1 , Ž c, 0. is a zero-divisor, so Ž c, 0. s Ž c, 0.Ž1, 0. s Ž0, 0., a contradiction unless c s 0. Hence, R1 ( ⺪ 2 . If R 2 is not an integral domain, then there is a nonzero b g ZŽ R 2 .. Then Ž1, b . is a zero-divisor of R which is not adjacent to Ž1, 0., a contradiction. Thus R 2 must be an integral domain. Note that if ZŽ R . is an annihilator ideal, then it is certainly maximal among annihilator ideals, and hence is prime. Ž¥. If R ( ⺪ 2 = A for A an integral domain, then Ž1, 0. is adjacent to every other vertex. If ZŽ R . s annŽ x . for some nonzero x g R, then x is adjacent to every other vertex. Example 2.1Ža. shows that both cases in Theorem 2.5 may occur for the same graph. However, if R is reduced and ⌫ Ž R . has a vertex adjacent to every other vertex, then R must have the form ⺪ 2 = A for A an integral domain. Note that the proof of Theorem 2.5 shows that if a vertex x of ⌫ Ž R . is adjacent to every other vertex, then either x is idempotent with Rx s 0, x 4
440
ANDERSON AND LIVINGSTON
a prime ideal of R, or ZŽ R . s annŽ x .. In fact, if ZŽ R . is an annihilator ideal, then annŽ ZŽ R ..* is precisely the set of vertices which are adjacent to every other vertex. Suppose that R is quasilocal with Žnonzero. maximal ideal M. If there is a least positive integer k with M k s 0, then ZŽ R . s M s annŽ x . for any nonzero x g M ky 1 Žso if < RrM < G 3, then there at least two vertices adjacent to every other vertex.. However, easy examples show that we need not have annŽ M . s M ky 1. Since ZŽ R . is always a union of prime ideals w9, p. 3x, ZŽ R . is a prime ideal if Žand only if. it is an ideal. If R is also Noetherian, then ZŽ R . is an annihilator ideal if and only if it is an Žprime. ideal w9, Theorems 6 and 82x. Recall that 04 is a primary ideal of R if and only if ZŽ R . s nilŽ R .. If dim R s 0, then ZŽ R . s nilŽ R . if and only if ZŽ R . is a Žprime. ideal of R; if R is finite, this is equivalent to R being local. COROLLARY 2.6. Let R be a commutati¨ e Noetherian ring. Then there is a ¨ ertex of ⌫ Ž R . which is adjacent to e¨ ery other ¨ ertex if and only if either
R ( ⺪ 2 = A, where A is an Ž Noetherian. integral domain, or ZŽ R . is an Ž prime. ideal of R. If in addition, dim R s 0, then either R ( ⺪ 2 = A, where A is a field, or 04 is a primary ideal of R Ž i.e., ZŽ R . s nilŽ R ... The Noetherian hypothesis is needed in Corollary 2.6. Let X 4 be an infinite family of indeterminates. Then R s ⺪ 2 w X 4xrŽ X2 4. is a nonNoetherian zero-dimensional quasilocal ring with maximal ideal ZŽ R . s nilŽ R . s Ž X 4.rŽ X2 4., but ZŽ R . is not an annihilator ideal. We next specialize to the case when R is finite.
COROLLARY 2.7. Let R be a finite commutati¨ e ring. Then there is a ¨ ertex of ⌫ Ž R . which is adjacent to e¨ ery other ¨ ertex if and only if either R ( ⺪ 2 = F, where F is a finite field, or R is local. Moreo¨ er, for some prime p and integer n G 1, < ⌫ Ž R .< s < F < s p n if R ( ⺪ 2 = F, and < ⌫ Ž R .< s p n y 1 if R is local.
We next determine when ⌫ Ž R . is a complete graph Ži.e., any two vertices are adjacent.. By definition, ⌫ Ž R . is complete if and only if xy s 0 for all distinct x, y g ZŽ R .. Except for the case when R ( ⺪ 2 = ⺪ 2 , our next theorem shows that we must also have x 2 s 0 for all x g ZŽ R . when ⌫ Ž R . is complete. So, except for that one case, nilpotent elements are detected by complete graphs. We denote the complete graph on n vertices by K n. THEOREM 2.8. Let R be a commutati¨ e ring. Then ⌫ Ž R . is complete if and only if either R ( ⺪ 2 = ⺪ 2 or xy s 0 for all x, y g ZŽ R .. Proof. Ž¥. By definition. Ž«. Suppose that ⌫ Ž R . is complete, but there is an x g ZŽ R . with x 2 / 0. We show that x 2 s x. If not, then x 3 s x 2 x s 0. Hence x 2 Ž x q x 2 . s 0 with x 2 / 0, so x q x 2 g ZŽ R .. If x q x 2 s x, then x 2 s 0, a contra-
ZERO-DIVISOR GRAPH
441
diction. Thus x q x 2 / x, so x 2 s x 2 q x 3 s x Ž x q x 2 . s 0 since ⌫ Ž R . is complete, again a contradiction. Hence x 2 s x. As in the proof of Theorem 2.5, we have R ( ⺪ 2 = A, and necessarily A ( ⺪ 2 . COROLLARY 2.9. Let R be a commutati¨ e ring. For x, y g ZŽ R ., define x ; y if xy s 0 or x s y, and define x ;* y if xy s 0. Ža. The relation ; is transiti¨ e Ž equi¨ alently, an equi¨ alence relation. if and only if ⌫ Ž R . is complete. Žb. The relation ;* is transiti¨ e Ž equi¨ alently, an equi¨ alence relation. if and only if ⌫ Ž R . is complete and R \ ⺪ 2 = ⺪ 2 . Proof. Both parts follow directly from Theorems 2.3 and 2.8. Suppose that R is a finite commutative ring and not isomorphic to ⺪ 2 = ⺪ 2 . By Corollary 2.7 and Theorem 2.8, ⌫ Ž R . is complete if and only if R is local with maximal ideal M and M 2 s 0. However, this does not hold if R is not finite. For example, if ZŽ R . 2 s 0, then by McCoy’s Theorem, also ZŽ Rw X x. 2 s 0. Hence ⌫ Ž⺪ 4 w X x. Žs 2⺪ 4 w X x y 04. is an infinite complete graph, but ⺪ 4 w X x is not local. Suppose that ⌫ Ž R . is complete and R \ ⺪ 2 = ⺪ 2 ; then by Theorem 2.8, ZŽ R . is an ideal of R with ZŽ R . 2 s 0. Hence ZŽ R . s nilŽ R . is a prime ideal of R. THEOREM 2.10. Let R be a finite commutati¨ e ring. If ⌫ Ž R . is complete, then either R ( ⺪ 2 = ⺪ 2 or R is local with char R s p or p 2 , and < ⌫ Ž R .< s p n y 1, where p is prime and n G 1. Proof. For a field F, the graph ⌫ Ž⺪ 2 = F . is not complete unless F s ⺪ 2 . So assume that R \ ⺪ 2 = ⺪ 2 . By Corollary 2.7, R must be local with maximal ideal M. Hence char R s p m for some prime p and integer m G 1. If m G 3, then R would have nonadjacent zero-divisors, a contradiction. Hence char R s p or p 2 . Since M is a p-group, < ⌫ Ž R .< s p n y 1 for some prime p and integer n G 1. We next show conversely that for each prime p and integer n G 1, there is a ring R such that ⌫ Ž R . is complete and < ⌫ Ž R .< s p n y 1. Thus K m may be realized as ⌫ Ž R . if and only if m s p n y 1 for some prime p and integer n G 1. Example 2.11 also shows that any graph ⌫ is a subgraph of a zero-divisor graph ⌫ Ž R . for some Žquasilocal. commutative ring R. EXAMPLE 2.11. Ža. Let p be prime, X s X␣ 4␣ g ᑜ a family of indeterminates, A s ⺪ p w X x, and I s Ž X␣ X < ␣ ,  g ᑜ 4.. Then R s ArI is a zero-dimensional quasilocal ring with maximal ideal M s ŽX.rI and M 2 s 0. Thus ⌫ Ž R . is complete. Moreover, < ⌫ Ž R .< s p n y 1 if < ᑜ < s n and < ⌫ Ž R .< s < ᑜ < if ᑜ is infinite. Also, for any field F, the zero-dimensional
442
ANDERSON AND LIVINGSTON
local ring S s F w X xrŽ X 2 . has a complete zero-divisor graph with < ⌫ Ž S .< s < F < y 1. Žb. Let p be prime and R s ⺪ p 2 . Then ⌫ Ž R . is complete with < ⌫ Ž R .< s p y 1. Note that ⺪ p 2 and ⺪ p w X xrŽ X 2 . are not isomorphic as rings, but have the same zero-divisor graph, namely, K py 1. In particular, none of ⺪ 2 = ⺪ 2 , ⺪ 9 , and ⺪ 3w X xrŽ X 2 . are isomorphic as rings, but all have the same complete zero-divisor graph K 2 . We next consider when ⌫ Ž R . has exactly one vertex which is adjacent to every other vertex. LEMMA 2.12. Let R be a finite commutati¨ e ring. If ⌫ Ž R . has exactly one ¨ ertex adjacent to e¨ ery other ¨ ertex and no other adjacent ¨ ertices, then either
R ( ⺪ 2 = F, where F is a finite field with < F < G 3, or R is local with maximal ideal M satisfying RrM ( ⺪ 2 , M 3 s 0, and < M 2 < F 2. Thus < ⌫ Ž R .< is either p n or 2 n y 1 for some prime p and integer n G 1. Proof. If R \ ⺪ 2 = F, then R is local with maximal ideal M by Corollary 2.7. Thus M s annŽ x . for a unique x g M. Let k be the least positive integer with M k s 0. Then M s annŽ y . for any nonzero y g M ky 1. Hence M ky1 s 0, x 4 , and thus < M ky 1rM k < s 2 yields RrM ( ⺪ 2 . If k G 4, then < M ky 2 < G 4. Hence for any distinct a, b g M ky 2 y M ky1 , ab g M 2 ky4 ; M k . Thus ab s 0, a contradiction. Hence M 3 s 0, and thus < M 2 < F 2. ŽNote that we may have M 2 s 0.. In Example 2.1Žd., we observed that ⌫ Ž⺪ 2 = F . is a star graph of order < F < when F is a finite field. We next use the previous lemma to show that, except for zero-divisor graphs of small order Žsee Example 2.1Ža.., this is the only way a star graph arises as ⌫ Ž R .. THEOREM 2.13. Let R be a finite commutati¨ e ring with < ⌫ Ž R .< G 4. Then ⌫ Ž R . is a star graph if and only if R ( ⺪ 2 = F, where F is a finite field. In particular, if ⌫ Ž R . is a star graph, then < ⌫ Ž R .< s p n for some prime p and integer n G 0. Con¨ ersely, each star graph of order p n can be realized as ⌫ Ž R ..
Proof. Suppose that ⌫ Ž R . is a star graph and R \ ⺪ 2 = F. By Corollary 2.7 and Lemma 2.12, we may assume that Ž R, M . is local with < M < s 2 k for some k G 3 Žsince M is a 2-group and < ⌫ Ž R .< G 4. and < M 2 < s 2. Let M s annŽ x . and choose distinct a, b, c, d g M* y x 4 . Then ab s ac s ad s x since M 2 s 0, x 4 and there are no other zero-divisor relations. Hence aŽ b y c . s aŽ b y d . s 0. Note that annŽ a. s 0, x 4 Ž a f annŽ a. since annŽ a. has even order. and thus b y c s b y d s x. Hence c s d, a contradiction. The ‘‘in particular’’ statement is clear and the ‘‘conversely’’ statement follows from the remarks preceding the theorem.
ZERO-DIVISOR GRAPH
443
EXAMPLE 2.14. For each integer n G 1, let R n s ⺪ 2 w X xrŽ X nq 1 . s ⺪ 2 w x x, a finite local ring. Then x n is the only vertex adjacent to every other vertex. However, for n G 3, ⌫ Ž R n . is not a star graph since the vertices x ny 1 q x n and x ny1 are also adjacent. Note that < ⌫ Ž R n .< s 2 n y 1.
3. AUTOMORPHISMS OF ⌫ Ž R . In this section, we study AutŽ ⌫ Ž R .., the group of graph automorphisms of ⌫ Ž R .. We first show that distinct ring automorphisms of R induce distinct graph automorphisms of ⌫ Ž R . when R is a finite ring which is not a field, and then we determine AutŽ ⌫ Ž R .. for several specific classes of finite rings. In particular, we compute AutŽ ⌫ Ž⺪ n ... Recall that a graph automorphism f of a graph ⌫ is a bijection f : ⌫ ª ⌫ which preserves adjacency. The set AutŽ ⌫ . of all graph automorphisms of ⌫ forms a group under the usual composition of functions. If < ⌫ < s n, then in the obvious way AutŽ ⌫ . is isomorphic to a subgroup of Sn , and clearly AutŽ K n . ( Sn . In fact, for a graph ⌫ of order n, AutŽ ⌫ . ( Sn if and only if ⌫ s K n. In the obvious way, each ring automorphism of R induces a graph automorphism of ⌫ Ž R .; we thus have a natural group homomorphism : AutŽ R . ª AutŽ ⌫ Ž R .. given by just restricting each f g AutŽ R . to ZŽ R .*. In general, AutŽ ⌫ Ž R .. is much larger than AutŽ R .; for example, each AutŽ⺪ n . is trivial. We first show that if R is a finite ring which is not a field, then : AutŽ R . ª AutŽ ⌫ Ž R .. is injective, i.e., an f g AutŽ R . is completely determined by its action on ZŽ R .. THEOREM 3.1. Let R be a finite commutati¨ e ring which is not a field and let f g AutŽ R .. If f Ž x . s x for all x g ZŽ R ., then f s 1 R . Thus : AutŽ R . ª AutŽ ⌫ Ž R .. is a monomorphism.
444
ANDERSON AND LIVINGSTON
Proof. First suppose that R has distinct maximal ideals M and N. Then M, N ; ZŽ R .. Since R s M q N, we must have f s 1 R . Thus we may assume that R is local with nonzero maximal ideal M Žs ZŽ R ... Let char R s p n and < RrM < s p m for some prime p and integers n, m G 1. We show that f Ž u. s u for each u g UŽ R .. Observe that for any m g M, um s f Ž um. s f Ž u. m, and hence Ž f Ž u. y u. m s 0. Thus f Ž u. s u q x for some x g M Žnote that x may depend on u. with xM s 0. Since < RrM < s m m p m , we have u p y1 s 1 q y for some y g M, and hence f fixes u p y1 . m m m m p my1 p y1 p y1 p y1 p y1 m . s f Ž u. Thus u s fŽu s Žu q x. su qŽp y m m 1.Ž u p y2 . x since xM s 0, and hence Ž p m y 1.Ž u p y2 . x s 0. Since p m y 1 m and u p y2 are units in R, we have x s 0. Thus f Ž u. s u for all u g UŽ R ., and hence f s 1 R . We next show that AutŽ ⌫ Ž⺪ n .. is a Žfinite. direct product of symmetric groups, and thus ⌫ Ž⺪ n . is highly symmetrical. The proof of this result gives the explicit construction of AutŽ ⌫ Ž⺪ n .; specific examples are given in parts Žc. and Žd. of Example 3.4. For any graph ⌫, the degree of a vertex x of ⌫ is ␦ Ž x . s < y g ⌫ < y is adjacent to x 4<. For a vertex x of a zero-divisor graph ⌫ Ž R ., we have ␦ Ž x . s
ZERO-DIVISOR GRAPH
445
Ž f < Vd ., with f < Vd viewed in the natural way as an element of Sn . By the d above comments, is a well-defined group monomorphism. To show that is surjective, it suffices to show that for each fixed d g X and permutation ␣ of Vd , there is an f g AutŽ ⌫ Ž⺪ n .. with f < Vd s ␣ and f < Vd⬘ s 1Vd⬘ for all d⬘ / d in X. This follows since for any x, y g Vd and a g ⺪ n , ax s 0 if and only if ay s 0. It is well known that any finite group G may be realized as AutŽ ⌫ . for some graph ⌫. It would be interesting to know exactly which groups may be realized as AutŽ ⌫ Ž R .. or AutŽ ⌫ Ž⺪ n ... The next corollary gives a partial answer for AutŽ ⌫ Ž⺪ n ... COROLLARY 3.3.
Let n G 4 be a nonprime integer. Then
Ža. AutŽ ⌫ Ž⺪ n .. is tri¨ ial if and only if n s 4. Žb. AutŽ ⌫ Ž⺪ n .. is abelian if and only if n s 4, 6, 8, 9, or 12. In particular, Aut Ž ⌫ Ž⺪ n .. ( ⺪ 2 when n s 6, 8, or 9, and AutŽ ⌫ Ž⺪12 .. ( ⺪ 2 = ⺪2 = ⺪2. Proof. The computations of AutŽ ⌫ Ž⺪ n .. for n s 4, 6, 8, and 9 follow easily from Theorem 3.2 or Example 2.1; for n s 12, see Example 3.4Žc.. We show that AutŽ ⌫ Ž⺪ n .. is nonabelian for all other nonprime integers n G 4. It is sufficient to show that in each case, some < Vd < G 3, and hence AutŽ ⌫ Ž⺪ n .. has a direct factor of S m for some m G 3. If n s 2 r with r G 4, then 2, 6, 104 ; V2 . If n s 3 r with r G 3, then 3, 6, 154 ; V3 . If n s p r with p G 5 prime, then p, 2 p, 3 p4 ; Vp . Let n s p1r 1 ⭈⭈⭈ pkr k be its prime factorization. If p1 G 3 and k G 2, then p1 , 2 p1, 4 p14 ; Vp . Thus we may assume that p1 s 2 with r s r 1 G 1 and 1 k G 2. If n ) 2 r ⭈ 3, then 2 r , 2 rq1 , 2 rq2 4 ; V2 r . Hence we may assume that n s 2 r ⭈ 3. If r G 3, then 2, 10, 144 ; V2 . Thus AutŽ ⌫ Ž⺪ n .. is abelian only if n s 4, 8, 9, 6, or 12. We conclude the paper with some specific examples of AutŽ ⌫ Ž R ... EXAMPLE 3.4. Ža. In Example 2.1Ža., we saw that the zero-divisor graphs of ⺖4 w X xrŽ X 2 . and ⺪ 3 = ⺪ 3 are a triangle and a square, respectively. Thus AutŽ ⌫ Ž⺖4 w X xrŽ X 2 ... ( S3 and AutŽ ⌫ Ž⺪ 3 = ⺪ 3 .. ( A 4 Ž( S2 wrS2 , where wr denotes the wreath product.. Žb. Let F and K be finite fields with distinct orders m and n, respectively. As in Example 2.1Žd., ⌫ Ž F = K . is the complete bipartite graph K my 1, ny1. Hence AutŽ ⌫ Ž F = K .. ( Smy 1 = Sny1 has order Ž m y 1.!Ž n y 1.!. Also, ⌫ Ž F = F . is the complete bipartite graph K my 1, my1. It may be shown that AutŽ K my 1, my1 . ( Smy 1wrS2 , which has order 2ŽŽ m y 1.!. 2 .
446
ANDERSON AND LIVINGSTON
Žc. We illustrate the proof of Theorem 3.2 by computing AutŽ ⌫ Ž⺪ 12 ... We have X s 2, 3, 4, 64 , and thus V2 s 2, 104 , V3 s 3, 94 , V4 s 4, 84 , and V6 s 64 . Hence AutŽ ⌫ Ž⺪12 .. ( S2 = S2 = S2 = S1 ( ⺪ 2 = ⺪ 2 = ⺪ 2 . This is also evident from the obvious symmetries of the zero-divisor graph of ⺪ 12 below.
Žd. For a more complicated example, let R s ⺪1225 Ž( ⺪ 25 = ⺪ 49 . with < ⌫ Ž R .< s 384. Then X s 5, 7, 25, 35, 49, 175, 2454 , and one may calculate that < V5 < s 168, < V7 < s 120, < V25 < s 42, < V35 < s 24, < V49 < s 20, < V175 < s 6, and < V245 < s 4. Thus AutŽ ⌫ Ž R .. ( S168 = S120 = S42 = S24 = S20 = S6 = S4 has order 168!⭈ 120!⭈ 42!⭈ 24!⭈ 6!⭈ 4!, which is approximately 6.192877 = 10 598. Note added in proof. In the paper, ‘‘Cycles of zero-divisors,’’ S. B. Mulay has shown, among other things, that Theorem 2.4 does indeed hold for any commutative ring.
ACKNOWLEDGMENTS Part of this paper is from w11x, which contains many more examples. We would like to thank Richard M. Stafford of the NSA, who also helped with calculations using Magma, and Aaron Lauve, who participated in an NSF-sponsored REU program with the first author at the University of Tennessee during the summer of 1998, for their many helpful discussions.
REFERENCES 1. D. D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra 159 Ž1993., 500᎐514.
ZERO-DIVISOR GRAPH
447
2. M. F. Atiyah and I. G. MacDonald, ‘‘Introduction to Commutative Algebra,’’ Addison᎐Wesley, Reading, MA, 1969. 3. I. Beck, Coloring of commutative rings, J. Algebra 116 Ž1988., 208᎐226. 4. B. Bollabas, ´ ‘‘Graph Theory, An Introductory Course,’’ Springer-Verlag, New York, 1979. 5. R. Diestel, ‘‘Graph Theory,’’ Springer-Verlag, New York, 1997. 6. N. Ganesan, Properties of rings with a finite number of zero-divisors, Math. Ann. 157 Ž1964., 215᎐218. 7. N. Ganesan, Properties of rings with a finite number of zero-divisors, II, Math. Ann. 161 Ž1965., 241᎐246. 8. F. Harary, ‘‘Graph Theory,’’ Addison᎐Wesley, Reading, MA, 1972. 9. I. Kaplansky, ‘‘Commutative Rings,’’ rev. ed., Univ. of Chicago Press, Chicago, 1974. 10. K. Koh, On ‘‘Properties of rings with a finite number of zero-divisors,’’ Math. Ann. 171 Ž1967., 79᎐80. 11. P. S. Livingston, ‘‘Structure in Zero-divisor Graphs of Commutative Rings,’’ Masters Thesis, The University of Tennessee, Knoxville, TN, December 1997. 12. B. R. McDonald, ‘‘Finite Rings with Identity,’’ Dekker, New York, 1974.