CUT VERTICES IN ZERO-DIVISOR GRAPHS OF FINITE COMMUTATIVE RINGS

CUT VERTICES IN ZERO-DIVISOR GRAPHS OF FINITE COMMUTATIVE RINGS M. AXTELL, N. BAETH, AND J. STICKLES Abstract. A cut vertex of a connected graph is a vertex whose removal would result in a graph having two or more connected components. We examine the presence of cut vertices in zero-divisor graphs of finite commutative rings and provide a partial classification of the rings in which they appear. 1. Introduction The concept of the graph of the zero-divisors of a commutative ring was first introduced by Beck in [5] when discussing the coloring of a commutative ring. In his work all elements of the ring were considered vertices of the graph. D.D. Anderson and Naseer used this same concept in [1]. We adopt the approach used by D.F. Anderson and Livingston in [2], now considered to be standard, and regard only nonzero zero-divisors as vertices of the graph. By a ring we mean a commutative ring with identity, typically denoted by R. We use R to denote the nonzero elements of R, U(R) to denote the units of R, Z(R) to denote the set of zero-divisors of R, and Z(R) to denote the set of nonzero zero-divisors of R. For any graph G, we denote the set of vertices of G by V (G). By the zero-divisor graph of R, denoted Γ(R), we mean the graph with V (Γ(R)) = Z(R), and for distinct r, s Z(R), there is an edge, denoted r s, connecting r and s if and only if rs = 0. If x 1, x 2,..., x n are vertices in Γ(R), then x 1 x 2 x n denotes a walk in Γ(R) from vertex x 1 to vertex x n, where x i is adjacent to x i+1 for 1 i n 1. If x 1, x 2,..., x n are distinct from one another, then the walk is called a path. A graph is said to be connected if there exists a path between any two distinct vertices. (For a general graph theory reference, see [6].) By [2, Theorem 2.4], Γ(R) is connected for all commutative rings R with identity. Zero-divisor graphs of commutative rings are typically studied to illuminate algebraic structure. With this in mind, we introduce cut vertices of Γ(R) and examine the algebraic information about a commutative ring R that can be gleaned when cut vertices are present. In Section 2 we define and give preliminary results about cut vertices in zero-divisor graphs of commutative rings. We then introduce our main goal: to understand when annihilator ideals corresponding to cut vertices in Γ(R) are properly maximal. In Section 3 we restrict our attention to finite rings and prove a series of results which allow us to focus only on finite local rings, which is the premise of Section 4. There we give a partial classification of which finite local rings have zero-divisor graphs with cut vertices. We conclude with a section introducing a generalization to sets of cut vertices. 2. Preliminary Results Given a graph G, let V (G) denote the set of vertices of G and E(G) the set of edges of G. A vertex a of a graph G is called a cut vertex if the removal of a and any edges incident on a creates a graph with more connected components than G. We say that G is split into two subgraphs X and Y via a if (1) V (X), V (Y ) 2, (2) X Y = G, (3) V (X) V (Y ) = {a}, (4) a is a cut vertex of G, and (5) x V (X)\{a} and y V (Y )\{a}, then x y is not in E(G). In particular, if Γ(R) is split into two subgraphs X and Y via a, then whenever x V (X)\{a} and y V (Y )\{a}, we have xy 0. Cut vertices were first examined in the context of zero-divisor graphs in [4] by a team of undergraduates at an NSF-sponsored REU. We provide a brief overview of their results below. Key words and phrases. commutative ring, cut vertex, zero-divisor graph. 1

2 M. AXTELL, N. BAETH, AND J. STICKLES Theorem 2.1. [4, Theorem 4.4] Let Γ(R) be split into two subgraphs X and Y via a. Then {0, a} is an ideal of R. Proof. Clearly, a a + a Z(R). Without loss of generality, suppose a + a V (X)\{a}. Let b V (Y )\{a} with ab = 0. Then b(a + a) = 0, a contradiction. So, a + a = 0 {0, a}. A similar argument shows that ra {0, a} for all r R. The converse of Theorem 2.1 is false; see Example 4.3. Theorem 2.1 implies that if x is a cut vertex of Γ(R), then rx = x for all r R\Z(R). If s / Z(R), then clearly s / Z(R). Thus, if r, s / Z(R), then x = rx = ( s)x. Hence, (r + s)x = 0, which implies that r + s Z(R). The proposition below summarizes this result. Proposition 2.2. If x is a cut vertex of Γ(R), then r + s Z(R) for all r, s / Z(R). However, it is not in general the case that if r + s Z(R) for all r, s / Z(R), then Γ(R) has a cut vertex. For example, consider Γ(Z 4 ). For a less trivial counterexample, see Example 4.3. Theorem 2.3. [4, Theorem 4.3] Let Γ(R) be split into two subgraphs X and Y via a such that V (X) > 2. If X is a complete subgraph, then b 2 = 0 for every b V (X). Proof. Let b V (X)\{a}. If b 2 = b, then b(b 1) = 0, implying b 1 Z(R) V (X). Let c V (X)\{a, b}. If c = b 1, then 0 = ac = a(b 1) = a, a contradiction. So, 0 = c(b 1) = cb c = c, giving c = 0, a contradiction. Hence, b 2 b, and since cb 2 = 0, we have b 2 V (X) {0}. Since X is complete, b(b 2 ) = 0. Also, since c(b 2 + b) = 0, it must be the case that b 2 + b V (X) {0}. Thus, 0 = b(b 2 + b) = b 2. Since c(a + b) = 0, we see that a + b V (X) {0}. Clearly, a + b a, and, since X is complete, 0 = a(a + b) = a 2. Example 2.4. The ring R = Z 2 Z 4 (see Figure 1) demonstrates the necessity of the condition that V (X) > 2 in Theorem 2.3. In the notation of the above proof, a = (0, 2), b = (1, 2), V (X) = {(0, 2), (1, 2)}, and V (Y ) = {(0, 2), (1, 0), (0, 1), (0, 3)}. We see that X is complete, but V (X) = 2. Notice that V (X)\{a, b} = and b 2 = (1, 0) V (Y ). Corollary 2.5. [4, Theorem 4.2] Let Γ(R) be split into two subgraphs X and Y via a such that V (X) > 2. If X is a complete subgraph, then V (X) {0} is an ideal. Proof. Choose b V (X)\{a}. By Theorem 2.3, we have bx = 0 = by for all x, y V (X) {0}. So, b(x + y) = 0, and hence, x + y V (X) {0}. Similarly, if r R, we have b(rx) = r(bx) = 0. Thus, rx V (X) {0}. The next example shows that the converse of Corollary 2.5 is false. Example 2.6. Let R = Z 2 Z 4. Observe that (1, 0) is a cut vertex of Γ(R) with V (X) = {(1, 0), (0, 2), (1, 2)} (see Figure 1). The set V (X) {(0, 0)} forms an ideal of R; however, the corresponding subgraph is not complete. Let a R. The set ann(a) = {r R ra = 0} is an ideal called the annihilator of a. If a 0, we say ann(a) is properly maximal if whenever b R \{a}, we have ann(a) ann(b). Proposition 2.7. If a is a cut vertex of Γ(R), then ann(a) is properly maximal. Proof. Suppose there exists b R \{a} such that ann(a) ann(b). Then whenever ad = 0, we have bd = 0. Let c, d V (Γ(R))\{a, b}. If a d is a path in Γ(R), then so is b d. Thus, if c a d is a path in Γ(R), then so is c b d. Hence, a is not a cut vertex of Γ(R). The converse of Proposition 2.7 is false, since ann(2) is properly maximal in Z 4, but 2 is not a cut vertex of Γ(Z 4 ). Less trivial examples are given below. Example 2.8. Consider R = Z 2 [X, Y ]/(X 3, Y 2 ). Let x = X and y = Y. Since x 2 y is the only nonzero element of R that annihilates both x and y, we see that ann(x 2 y) is properly maximal in R. However, the set {x 2, xy, x 2 + xy} induces a complete subgraph of Γ(R), and it is straightforward to verify that every element in Z(R) annihilates some element of this set. Hence, the subgraph induced by V (Γ(R))\{x 2 y} is connected, and thus, x 2 y is not a cut vertex of Γ(R).

CUT VERTICES IN ZERO-DIVISOR GRAPHS OF FINITE COMMUTATIVE RINGS 3 (1,2) (0,2) (1,0) (0,1) (0,3) Figure 1. Γ(Z 2 Z 4 ) Example 2.9. Consider R = Z 2 [X, Y 1, Y 2,...]/(X 2, {XY i } i Z +, {Y i Y j } i,j Z +,i j). Clearly, if x = X, then ann(x) is properly maximal. But x is not a cut vertex, since given f, g Z(R), there exists a j such that y j g = y j f = 0, where y j = Y j. Note that if R is a Noetherian ring with the property that ann(a) ann(b) whenever a b, then Γ(R) has only finitely many cut vertices by Proposition 2.7 [8, Theorems 6 and 80]. 3. Finite Rings and the Non-local case For the rest of the paper, we assume that R is a finite commutative ring with identity. Thus, any element of R is either a unit or a zero-divisor. Recall that since R is finite, R can be decomposed into a direct product R = R 1 R 2 R n, where each R i is a nontrivial finite local ring (cf. [9, Theorem VI.2]). Lemma 3.1. If R is a finite commutative ring that is not local and a is a cut vertex of Γ(R), then a = (a 1, a 2,..., a n ) has exactly one nonzero component. Proof. Without loss of generality, suppose a = (a 1, a 2,..., a n ) is a cut vertex of Γ(R), where a 1, a 2 0. Then ann((a 1, a 2,..., a n )) ann((a 1, 0,..., 0)), contradicting Proposition 2.7. The converse of Lemma 3.1 is false, since (0, 1) is not a cut vertex in Z 2 Z 4 as seen in Figure 1. Lemma 3.2. Suppose R is a finite, non-local commutative ring not isomorphic to Z 2 Z 2. Then a vertex of Γ(R) is a cut vertex with unit nonzero component if and only if it has precisely one nonzero component, and the ring corresponding to its nonzero component is isomorphic to Z 2. Proof. ( ) Let a be a cut vertex of Γ(R). Without loss of generality, assume a = (u, 0,..., 0), where u U(R 1 ). If there exists b R 1 \{0, u}, then ann((u, 0,..., 0)) ann((b, 0,..., 0)), contradicting Proposition 2.7. Thus, R 1 = Z2. ( ) Without loss of generality, let R = Z 2 S with S > 2. We have ann((0, 1)) = {(0, 0), (1, 0)}, so (1, 0) is a cut vertex of Γ(R).

4 M. AXTELL, N. BAETH, AND J. STICKLES Lemma 3.3. Let R be a ring, R Z 2 Z 2, and a Z(R). If Z(R) 2 and a is not a cut vertex of Γ(R), then (a, 0) is not a cut vertex of Γ(R S) for any ring S. Proof. Let (b, x) V (Γ(R S)) with (a, 0) (b, x). It suffices to show that there is a walk from (b, x) to (a, 1) in Γ(R S) that does not contain (a, 0). Since Z(R) 2, we can choose a c Z(R) \{a} with ac = 0. If b {0, a}, then (b, x) (c, 0) (a, 1) is a walk in Γ(R S). Otherwise, a b is an edge in Γ(R). If Z(R) {a, b}, then, since a is not a cut vertex of Γ(R), there exists d Z(R) \{a, b} with bd = 0. Then (b, x) (d, 0) (b, 0) (a, 1) is a walk in Γ(R S). If Z(R) = {a, b}, then (b, x) (b, 0) (a, 1) is a walk if x 0 [2, Theorem 2.8], and (b, x) (a, 1) is a walk if x = 0. Thus, (a, 0) is not a cut vertex of Γ(R S). Example 2.6 shows the hypothesis that Z(R) 2 is necessary, since 2 is not a cut vertex of Γ(Z 4 ), but (0, 2) is a cut vertex of Γ(Z 2 Z 4 ). The following example shows that the converse of this lemma is false. Example 3.4. Let R = Z 4 [X]/(X 2 ) and denote x = X. As seen in the proof of the next lemma, (2x, 0) is not a cut vertex of Γ(R S) for any ring S. However, as seen in Figure 2, 2x is a cut vertex in Γ(R). 2 2 + 2x x 2x 2 + x 3x 2 + 3x Figure 2. Γ(Z 4 [X]/(X 2 )) The following lemma provides a partial converse to Lemma 3.3. Lemma 3.5. Let R 1 and R 2 be finite commutative rings such that Γ(R 1 ) is split into two subgraphs X and Y via a 1. Then (a 1, 0) is a cut vertex of Γ(R 1 R 2 ) if and only if one of X or Y contains at least one isolated vertex when a 1 and its incident edges are removed. Proof. ( ) Suppose neither X nor Y contains an isolated vertex when a 1 and its incident edges are removed. Let (b, x) V (Γ(R 1 R 2 )) with (a 1, 0) (b, x) an edge in Γ(R 1 R 2 ). It suffices to show that there is a walk from (b, x) to (a 1, 1) in Γ(R 1 R 2 ) that does not contain (a 1, 0). If b {0, a 1 }, then choose any c Z(R 1 ) \{a 1 } with a 1 c = 0. Then (b, x) (c, 0) (a 1, 1) is a walk in Γ(R 1 R 2 ). Otherwise, a 1 b is an edge in Γ(R 1 ). Since b is not isolated when a 1 and its incident edges are removed from Γ(R 1 ), there exists d Z(R 1 ) \{a 1, b} with bd = 0. Then (b, x) (d, 0) (b, 0) (a 1, 1) is a walk in Γ(R 1 R 2 ). Thus, (a 1, 0) is not a cut vertex of Γ(R 1 R 2 ). ( ) Let b R 1 be an isolated vertex when a 1 and its incident edges are removed from Γ(R 1 ). Since (a 1 + b)b = b 2 and b is isolated, we see that b 2 0. This implies ann(b) = {0, a 1 }, so ann((b, 1)) = {(0, 0), (a 1, 0)}. Hence, (a 1, 0) is a cut vertex of Γ(R 1 R 2 ).

CUT VERTICES IN ZERO-DIVISOR GRAPHS OF FINITE COMMUTATIVE RINGS 5 Theorem 3.6. Let R = R 1 R 2 R n with each R i a finite local ring and n 2. If R Z 2 Z 2, then a vertex a = (a 1, 0,..., 0) R is a cut vertex of Γ(R) if and only if one of the following holds. (1) a 1 = 1 with R 1 = Z2. (2) R 1 = Z4 or Z 2 [X]/(X 2 ) with a 1 = 2 or X. (3) a 1 is a cut vertex in Γ(R 1 ) that isolates at least one vertex when removed. Proof. ( ) Suppose (a 1, 0,..., 0) is a cut vertex of Γ(R). If a 1 U(R 1 ), then R 1 = Z2 and a 1 = 1 by Lemma 3.2. So, suppose a 1 Z(R 1 ). By Lemma 3.3, either Z(R 1 ) < 2 or a 1 is a cut vertex of R 1. By assumption, Z(R 1 ). If Z(R 1 ) < 2, then R 1 = Z4 or Z 2 [X]/(X 2 ) with a 1 = 2 or X by [2]. If a 1 is a cut vertex of Γ(R 1 ), then a 1 is a cut vertex in Γ(R 1 ) that isolates at least one vertex when removed by Lemma 3.5. ( ) By Lemmas 3.2 and 3.5, we only need to show that if R 1 = Z4 or Z 2 [X]/(X 2 ) with a 1 = 2 or X, then (a 1, 0,..., 0) is a cut vertex of Γ(R). It is clear that in both of these cases we have ann ((a 1, 1,..., 1)) = {(0, 0,..., 0), (a 1, 0,..., 0)}. Hence, (a 1, 0,..., 0) is a cut vertex of Γ(R). Theorem 3.6 also shows that cut vertices of zero-divisor graphs of direct products always isolate vertices. If R 1 = Z2 and S is not isomorphic to Z 2, then the cut vertex (1, 0) of Γ(R 1 S) isolates (0, 1). Similarly, (2, 0) (resp. (x, 0)) isolates (2, 1) (resp. (x, 1)) in Γ(Z 4 S) (resp. Γ(Z 2 [X]/(X 2 ) S)). Thus, we get the following corollary. Corollary 3.7. Let R be a finite commutative ring. If Γ(R) contains a cut vertex that does not isolate any vertices, then R is a local ring. The converse to Corollary 3.7 is false. In Γ(Z 8 ), 4 is the only cut vertex, and it isolates 2. If x is a cut vertex of Γ(R) such that x 2 0, then by Theorem 2.1, x 2 = x. Hence, R is not local. Thus, R = R 1 R 2 R n with each R i a local ring. Also, if x = (a 1, 0, 0,..., 0), then a 2 1 = a 1, which implies that R 1 = Z2 and a 1 = 1. 4. Finite Local Rings The previous section has reduced the classification of the cut vertices of graphs of decomposable rings to classifying cut vertices of graphs of local rings. In this section, we assume that R is a finite local ring with unique maximal ideal M. In this case we have M = Z(R) and M is nilpotent. Throughout this section we also assume that k is the least positive integer such that M k = 0. Theorem 4.1. Let R be a finite local ring with maximal ideal M. M k 1 = {x, 0}. If x is a cut vertex of Γ(R), then Proof. Suppose x / M k 1. If 0 r M k 1, then ar = 0 for all a Z(R). Thus, a r is an edge in Γ(R) for all a Z(R) \{r}. Thus, x is not a cut vertex of Γ(R). By an identical argument, if x is a cut vertex of Γ(R), then no other nonzero element is in M k 1. Corollary 4.2. Let R be a finite local ring. Then x is a cut vertex of Γ(R) if and only if x is the unique cut vertex of Γ(R). The converse to Theorem 4.1 is false. Example 4.3. Let R = Z 8 [X]/(2X, X 2 ). Observe that M = Z(R) = {0, 2, 4, 6, X, X + 2, X + 4, X + 6} and M 3 = {0}. Though M 2 = {0, 4}, we see that 4 is not a cut vertex of Γ(R) since M = ann(4) = ann(x) = ann(x + 4). See Figure 3. By Theorems 2.1 and 4.1, if x is a cut vertex of a finite local ring R, then x 2 = 0. So, in the non-local case, by Theorem 3.6, if x is a cut vertex such that x 2 0, then Z 2 is a factor in the ring decomposition. Further, if R Z 2 Z 2, then the number of cut vertices with x 2 0 corresponds to the number of copies of Z 2 in the decomposition. Theorem 4.4. Let R be a finite local ring with maximal ideal M. R = 2 n+1 and M = 2 n for some n Z +. If x is a cut vertex of Γ(R), then

6 M. AXTELL, N. BAETH, AND J. STICKLES 2 6 4 4 + x 2 + x x 6 + x Figure 3. Γ(Z 8 [X]/(2X, X 2 )) Proof. We have that M k = {0} and M k 1 = {0, x} by Theorem 4.1. Thus, M k 1 /M k = 2. Since M k 1 /M k is an R/M-vector space, and since M k 1 /M k = 2, we see that R/M = 2. Similarly, M i /M i+1 is an R/Mvector space, so M i /M i+1 is a power of 2 for all 1 i k. Hence, R = R/M M/M 2 M k 1 /M k = 2 n+1 for some n. Thus, R = 2 n+1 and M = 2 n. Corollary 4.5. Let R be a finite local ring. If x is a cut vertex of Γ(R), then R has characteristic 2 m. As shown by Example 4.3, the converse of Corollary 4.5 is false. Recall that a principal ideal ring (PIR) is a ring in which all ideals are principal, and a special principal ideal ring (SPIR) is a PIR with a unique prime ideal, and this prime ideal is nilpotent. Clearly, an SPIR is also local. If R is a finite local PIR, then R is an SPIR. To see this, recall that if every nonunit in a ring R is nilpotent, then R has a unique prime ideal, namely the nilradical of R. It is also of interest to note that if R is a commutative ring with identity, then R has only finitely many principal ideals if and only if R is a finite direct product of SPIR s and finite local rings [3, Lemma 3.4]. Thus, the decomposition of a finite ring into local rings will often involve SPIR s. If R is a finite SPIR, then R is local with maximal ideal M and Z(R) = M = (a). Clearly, M i = (a i ). In the SPIR case, the converse of Theorem 4.1 is true. Theorem 4.6. Let R be a finite SPIR with maximal ideal M = (a). Let x = a k 1, where k 3 is the minimal positive integer such that a k = 0. The following are equivalent. (1) x is a cut vertex of Γ(R). (2) M k 1 = {0, x}. (3) R/M = Z 2. (4) ua k 1 = a k 1 for all u U(R). (5) ann(x) is properly maximal. Proof. (1) (2) Theorem 4.1.

CUT VERTICES IN ZERO-DIVISOR GRAPHS OF FINITE COMMUTATIVE RINGS 7 (2) (3) If M k 1 = {0, x}, then as in the proof of Theorem 4.4, M k 1 is an R/M-vector space with R/M = 2. (3) (4) If R/M = Z 2, then every u U(R) can be written as u = 1 + ra for some r R. Hence, ua k 1 = (1 + ra)a k 1 = a k 1. (4) (5) Clearly, (a) = M = ann(x), and thus, ann(x) is maximal. If there exists a b Z(R) \{x} such that ann(b) = M, then b = ua k 1, where u is a unit unique modulo (a) by [9]. By hypothesis, b = ua k 1 = a k 1 = x. Thus, ann(x) is properly maximal. (5) (1) Since M k = 0, it follows that ann(y) = M for every y M k 1. Thus, ann(x) = M, and since ann(x) is properly maximal, M k 1 = {0, x}. We claim that ann(a) = M k 1 = (a k 1 ) = (x). Clearly, M k 1 ann(a). For the other containment, let 0 y ann(a). From [9], y = ua j, where j is unique and u is a unit unique modulo (a k j ). Thus, we have 0 = ya = ua j a. This implies that a j+1 = 0, and hence, j = k 1. Thus, y (a k 1 ) = M k 1. This implies that the only nonzero annihilator of a is a k 1 = x. Thus, x is a cut vertex. The rings Z 4 and Z 2 [X]/(X 2 ) show that k 3 is a necessary hypothesis. Recall that GR(p n, r) denotes the Galois extension of Z p n of degree r and is referred to as a Galois ring. Also, in the Galois ring Z p n, a polynomial g(x) = X s + p(a s 1 X s 1 + + a 1 X + a 0 ) Z p n[x], where a 0 U(Z p n), is called an Eisenstein polynomial over Z p n (cf. [9, p. 342]). Corollary 4.7. Let R be a finite SPIR and let k 3. Then Γ(R) has a cut vertex if and only if R = Z 2 n[x]/(g(x), 2 n 1 X t ), where t = k (n 1)s > 0 and g(x) is an Eisenstein polynomial of degree s over Z 2 n. Proof. ( ) By [9, Theorem XVII.5], R = GR(p n, r)[x]/(g(x), p n 1 X t ), where t = k (n 1)s > 0 and g(x) is an Eisenstein polynomial of degree s over GR(p n, r). Since Γ(R) has a cut vertex, R/M = Z 2 by Theorem 4.6. Thus, by the remark preceeding Lemma XVII.7 of [9], R = Z 2 n[x]/(g(x), 2 n 1 X t ) where t = k (n 1)s > 0 and g(x) is an Eisenstein polynomial of degree s over Z 2 n. ( ) Assume R = Z 2 n[x]/(g(x), 2 n 1 X t ), where t = k (n 1)s > 0 and g(x) is an Eisenstein polynomial of degree s over Z 2 n. By the remark preceeding Lemma XVII.7 of [9], R/M = Z 2, and thus, Γ(R) has a cut vertex by Theorem 4.6. Note that Theorem 4.6 also shows that the converse of Proposition 2.7 holds when R is a finite SPIR other than Z 4 or Z 2 [X]/(X 2 ). In addition, cut vertices of SPIR s will isolate vertices. Not all SPIR s have cut vertices as can be seen by examining F 4 [X]/(X 3 ). In this ring, the maximal ideal M has M 3 = 0 and M 2 = 4; hence, F4 [X]/(X 3 ) has no cut vertices by Theorem 4.6. 5. Cut-Sets of Γ(R) In the summer of 2009, a group of REU students generalized the idea of a cut vertex to that of a cut-set. We include their definition and a preliminary result below, both of which can be found in [7]. Definition 5.1. A set A Z(R), where Z(R) = Z(R)\{0}, is said to be a cut-set if there exist c, d Z(R) \A where c d such that every path in Γ(R) from c to d involves at least one element of A, and no proper subset of A satisfies the same condition. Another way to define a cut-set is as a set of vertices {a 1, a 2, a 3...} in a connected graph G where G can be expressed as a union of two subgraphs X and Y such that E(X), E(Y ), E(X) E(Y ) = E(G), V (X) V (Y ) = V (G), V (X) V (Y ) = {a 1, a 2, a 3...}, V (X)\{a 1, a 2, a 3...}, V (Y )\{a 1, a 2, a 3...}, and no proper subset of {a 1, a 2, a 3...} also acts as a cut-set for any choice of X and Y. A cut vertex can be thought of as a cut-set with only one element. For a cut-set A in Γ(R), a vertex a / A is said to be isolated, or an isolated point, if ann(a) A. Example 5.2. Consider Γ(Z 12 ) shown in Figure 4. In this graph, 6 is a cut vertex, and {4, 8} is a cut-set isolating 3 and 9. The main result of [7] is the following theorem. Theorem 5.3. [7, Theorem 3.7]A set A of V (Γ( m i=1 Z n i )) with m 2 is a cut-set if and only if A = {(0, 0,..., a i1,..., 0), (0, 0,..., a i2,..., 0),..., (0, 0,..., a ik,..., 0)} where {a i1, a i2,..., a ik } = ann(p)\{0} for some prime p Z such that p n i in Z ni.

8 M. AXTELL, N. BAETH, AND J. STICKLES 3 4 9 8 6 2 10 Figure 4. Γ(Z 12 ) Given the interesting ring-structure results that examining the cut vertices of Γ(R) led to, it is hoped that further inquiry into cut-sets will expand these results. References [1] Anderson, D.D. and Naseer, M., Beck s coloring of a commutative ring, J. Algebra 159 (1993), 500-514. [2] Anderson, D.F. and Livingston, P., The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434-447. [3] Axtell, M., U-factorizations in commutative rings with zero divisors, Comm. Alg. 30(3) (2002), 1241-1255. [4] Axtell, M., Stickles, J., Trampbachls, W., Zero-divisor ideals and realizable zero-divisor graphs, Involve 2(1) (2009), 17-27. [5] Beck, I., Coloring of commutative rings, J. Algebra 116 (1988), 208-226. [6] Chartland, G., Introductory Graph Theory, Dover Publications, Inc., New York, 1977. m [7] Cote, B., Ewing, C., Huhn, M., Plaut, C.M., Weber, D., Cut-Sets and Cut Vertices in Γ( Z ni ), submitted. i=1 [8] Kaplansky, I., Commutative Rings, Polygonal Publishing House, Washington, New Jersey, 1974. [9] McDonald, B., Finite rings with identity, Marcel Dekker, Inc., New York, 1974. [10] Redmond, S., On zero-divisor graphs of small finite commutative rings, Discrete Math, 307 (9) (2007), 1155-1166. Department of Mathematics, University of St. Thomas, St. Paul, MN 55105 E-mail address: maxtell@stthomas.edu Department of Mathematics and Computer Science, University of Central Missouri, Warrensburg, MO 64093 E-mail address: baeth@ucmo.edu Department of Mathematics, Millikin University, Decatur, IL 62522 E-mail address: jstickles@millikin.edu