Irreducible Divisor Graphs

Irreducible Divisor Graphs Jim Coykendall Department of Mathematics North Dakota State University Fargo, ND 58105-5075 Jack Maney Department of Mathematical Sciences The University of South Dakota 414 E. Clark St. Vermillion, SD 57069 October 21, 2005 Abstract In [6], I. Beck introduces the idea of a zero-divisor graph of a commutative ring. We generalize this idea to study factorization in integral domains and define irreducible divisor graphs. We use these irreducible divisor graphs to characterize certain classes of domains, including UFDs.

1

Introduction

In this paper, R will denote an integral domain (commutative, with nonzero identity). Often, but not always, we will also assume that R is an atomic domain (that is, every nonzero nonunit element of R can be written as a product of irreducibles). If S is a subset of R, we denote S \ {0} by S ∗ . Also, N and Fq denote the natural numbers and the field of q elements, respectively. If G = (V, E) is a graph and if v is a vertex of G, then by v ∈ G we will mean v ∈ V . Also, V and E may also be respectively denoted by V (G) and E(G) if there is any danger of confusion as to which graph is being referred. A recent subject of study linking commutative ring theory with graph theory has been the concept of the zero-divisor graph of a commutative ring. Let R be a commutative ring with identity and denote the set of (nonzero) zero-divisors of R as Z(R). The zero-divisor graph of R, denoted by Γ(R), is the graph with vertex set Z(R) and if a, b ∈ Z(R), ab ∈ E(Γ(R)) if and only if ab = 0. Zero-divisor graphs were introduced by I. Beck in [6], and they have been studied and generalized even more recently (cf. [5], [3], [11], [8], [10], [14], [4],

1

[15], [1], [2], [12], [16], [9], and [17]). In this paper, we stretch this idea in a different direction, applying it to factorization in integral domains. If R is atomic, it is well-known that nonzero nonunit elements of R need not factor uniquely into irreducibles. If a nonzero nonunit x ∈ R (with R atomic) does not factor uniquely into irreducibles, then it follows that there exist two nonassociate irreducible divisors, π and ξ of x such that x = πα1 α2 · · · αm = ξβ1 β2 · · · βm , with αi , βj irreducible, and with π nonassociate to each βj and likewise with ξ and each αi . When this happens, we will say that π and ξ show up in two different factorizations of x. Equivalently, if it is not the case that any two nonassociate irreducible divisors of x show up in two different factorizations of x, then x factors uniquely into irreducibles. Thus, if x is a nonzero nonunit element of an (atomic) domain R, then one approach to studying the factorization(s) of this (fixed) x lies in looking at how many pairs of nonassociate irreducible divisors of x are both in a factorization of x (or, in other words, looking at how many pairs of nonassociate irreducible divisors of x have their product dividing x). This leads to the development of the irreducible divisor graph. If R is a domain, we denote the set of irreducibles of R by Irr(R). However, as is the case when studying factorization in integral domains, we will generally not distinguish between an irreducible and its associates. So, we define Irr(R) to be a (pre-chosen) set of coset representatives, one representative from each coset in the collection {πU (R) | π ∈ Irr(R)}. We also recall that a graph G (with no loops or multiple edges) is a clique if G is a complete graph, that is, if there is an edge between every pair of vertices. A clique on n vertices (n ∈ N) will be denoted by Kn . If G is a graph with a subgraph H, we recall that H is an induced subgraph of G if for all u, v ∈ H, uv ∈ E(G) implies that uv ∈ E(H). For v ∈ G, we denote the neighborhood of v (in G) by N (v) = NG (v) := {u ∈ G | uv ∈ E(G)}. We will need a new graph-theoretic definition that we will find helpful. Definition 1.1 Let G be a graph. We say that G is a pseudo-clique if G is a complete graph having some number (possibly zero) of loops. It immediately follows that a clique is a pseudo-clique. Definition 1.2 Let R be a domain, and let x ∈ R be a nonzero nonunit that can be factored into irreducibles. 1. We define the irreducible divisor graph of x to be the graph G(x) = (V, E) where V = {y ∈ Irr(R) | y|x}, and given y1 , y2 ∈ Irr(R), y1 y2 ∈ E if and only if y1 y2 |x. Further, we attach n − 1 loops to the vertex y if y n divides x.

2

2. We define the reduced divisor graph of x to be the subgraph of G(x) containing no loops, and we denote the reduced divisor graph of x as G(x). 3. If R is an antimatter domain, that is, a domain having no irreducibles (and this includes the class of fields), then we define G(x) = G(x) to be the empty graph for each x ∈ R. 4. If A ⊆ V (G(x)), then by GA (x), we mean the induced subgraph of G(x) on N (A). If A = {π1 , π2 , · · · , πn }, then GA (x) will be denoted by G(π1 ,π2 ,··· ,πn ) (x) and if A = {π}, then GA (x) will be denoted by Gπ (x). Note that in G(x) it is possible for π to be in N (π). In fact, this happens precisely when π 2 |x. We will mainly be interested in the case of R being atomic. Throughout the rest of this paper, we will assume that R is not an antimatter domain.

2

Examples

√ Example 2.1 Let R = Z[ −5]. Using norms, it is easy to see that the only nonassociate irreducible factorizations of 6 are: √ √ 6 = 2 · 3 = (1 − −5)(1 + −5). So, G(6) is as follows: •2 •

• 1+

3

•

√ −5

√ 1− −5

Also, the only irreducible factorizations of 18 in R are: √ √ √ √ 18 = 2 · 32 = 3(1 − −5)(1 + −5) = 2(2 + −5)(2 − −5). So, G(18) is as follows:

2

ÄÄ ÄÄ Ä ÄÄ ÄÄ Ä Ä ÄÄ

• √ 2+ −5

3

•

•? ?? ?? ?? ?? ?? ?? • • √ √

• √ 2− −5

1− −5

1+ −5

√ Example 2.2 Let R = Z[ −14]. Then, again using norms, we see that the only irreducible factorizations of 81 are: √ √ 81 = 34 = (5 − 2 −14)(5 + 2 −14). 3

Thus G(81) is as follows: √

3

• 5+2

−14

• 5−2

√ −14

•

Example 2.3 Let R be a UFD, and let x be any nonzero nonunit. Then we may factor x as pa1 1 pa2 2 · · · pann where ai ∈ N and the pi are distinct nonassociate primes. Since this is the only way to factor x into irreducibles, we see that G(x) is isomorphic to Kn with ai − 1 loops on each vertex pi . It also follows that G(x) ∼ = Kn . Example 2.4 Let R = Q[x2 , x3 ] and let f (x) = x9 − x10 . The only irreducible factorizations of f in R are: f (x) = x2 · x2 · x3 (x2 − x3 ) = x2 · x2 · x2 (x3 − x4 ) = x3 · x3 (x3 − x4 ). And G(f ) is as follows:

x2 • ? ÄÄ ??? Ä ?? Ä ?? ÄÄ Ä 3 4 Ä x −x x2 −x3 •? • ?? ÄÄ ?? ÄÄ ?? Ä ? ÄÄÄ • x3

Example 2.5 Let F = F2 , and let y be a root of the (irreducible) polynomial X 3 + X + 1 ∈ F [x]. Let K = F [y] and let V be the F -subspace of K with basis {1, y}. Finally, let R = F + xV + x2 K[[x]]. After some calculations, it is apparent that V = {0, 1, y, y 3 } and that Irr(R) = {x, yx, y 3 x, y 5 x2 }. Furthermore, the only nonassociate irreducible factorizations of x5 are: (x)5 = (y 5 x2 )(y 3 x)3 = (y 5 x2 )(yx)2 x = x2 (y 3 x)2 (yx) = (yx)4 (y 3 x).

4

Therefore G(x5 ) is as follows: x• Ä ??? Ä ?? ÄÄ ?? ÄÄ ? y 5 x2 Ä 3 y x Ä •? • ?? Ä Ä ?? Ä ?? ÄÄ ? ÄÄÄ • yx

and G(x5 ) has four loops on x, two loops on y 3 x, and three loops on yx. We now notice a contrast between these examples. In Examples 2.1, 2.2, and 2.3 every (maximal) pseudo-clique corresponds to a factorization. For √ example, in the second graph in 2.1, we have a triangle formed by 2 and 2 ± −5 and we √ √ also have 2(2 + −5)(2 − −5) = 18. Similarly, in Example 2.2, the vertex 3 (along√with its 3 loops) makes a factorization of 81, as does the K2 formed by 5 ± 2 −14. These examples are “nice” in the sense that it is easy to pick out the irreducible factorizations just by looking at the irreducible divisor graph. In contrast, in Example 2.4, we have a triangle formed by x2 , x3 , and x3 −x4 . This triangle also forms a maximal pseudo-clique (since G(f ) is not a psuedoclique). However, x2 · x3 · (x3 − x4 ) does not divide f (x). Also, in Example 2.5, G(x5 ) is a pseudo-clique. However, y 9 x5 = y 2 x5 = x(yx)(y 3 x)(y 5 x2 ) does not divide x5 in R. The phenomenon behind this contrast will be studied in the sequel.

3

Preliminary Results

Recall that R is a Finite Factorization Domain (FFD) if R is atomic and every nonzero nonunit element of R has finitely many nonassociate irreducible divisors. Proposition 3.1 Let R be an atomic domain. Then R is an FFD if and only if G(x) is finite for each nonzero nonunit x ∈ R. Proof: If R is an FFD, then every nonzero nonunit x ∈ R has but finitely many nonassociate irreducible factors, whence the vertex set of G(x) is finite. On the other hand, if G(x) is finite for each x, then each x has only finitely many nonassociate irreducible divisors, and R is an FFD. ¥ Lemma 3.2 Let R be a domain and let x ∈ R be a nonzero nonunit that can be factored into irreducibles, and let π be an irreducible divisor of x. Then the vertex sets of Gπ (x) and G( πx ) coincide. In other words, the vertices of G( πx ) are precisely those in the neighborhood of π. What is more, the edge set of G( πx ) is contained in the edge set of Gπ (x). Proof: Note that the result is vacuously true if x ∈ Irr(R).

5

Assume that there exists another irreducible divisor of x that is adjacent to π, call it ξ. Since ξ and π are adjacent in G(x), we must have that ξπ|x whence ξ ∈ G( πx ). Conversely, if ξ is a vertex of G( πx ), then ξ is an irreducible divisor of πx whence ξπ|x and ξ is a vertex in Gπ (x). Now, let α, β ∈ G( πx ) have an edge between them. Then αβ| πx , yielding that αβπ|x Thus αβ|x combined with the fact that α, β ∈ N (π) yields that αβ ∈ E(Gπ (x)). If no other irreducible divisors of x are adjacent to π, then (up to a unit of R), we have x = π k for some k ≥ 2. It follows that π is adjacent to itself in G(x), whence it is easy to see that the assertions follow. ¥ We recall that an atomic domain R is a half factorial domain, or HFD, if given any collection of irreducibles {α1 , α2 , · · · , αm , β1 , β2 , · · · , βn } with α1 · · · αm = β1 · · · βn , then m = n. Theorem 3.3 Let R be an atomic domain. Then the following are equivalent: 1. R is an HFD. 2. For any nonzero nonunit x of R, and for any irreducible factorization π1a1 π2a2 · · · πnan with the πi ’s pairwise nonassociate, the sum of the number of vertices and the number of loops in G(π1 ,π2 ,··· ,πn ) (x) is constant. Proof: (1 ⇒ 2): Clear from the definition of HFD. (2 ⇒ 1): Let bm π1a1 π2a2 · · · πnan = ξ1b1 ξ2b2 · · · ξm be two irreducible factorizations of x. In G(π1 ,π2 ,··· ,πn ) (x) (resp. G(ξ1 ,ξ2 ,··· ,ξn ) (x)), πi (ξj ) has ai − 1 (bj − 1) loops. Thus, by hypothesis, n+

n X

m n m X X X (ai − 1) = m + (bj − 1) ⇒ ai = bj ,

i=1

j=1

i=1

j=1

and R is an HFD. ¥

4

Boundary Valuation Domains

We will now characterize Boundary Valuation Domains (which were defined and studied by the second author in [13]) by their irreducible divisor graphs. We first require some definitions. Definition 4.1 Let R be an HFD with quotient field K. If R 6= K, we define π1 π2 · · · πt the boundary map ∂R : K ∗ −→ Z by ∂R (α) = t − s where α = where δ1 δ2 · · · δs πi , δj ∈ Irr(R) for each i and j. If R = K, then we declare ∂R (α) = 0 for all α ∈ K ∗ . The boundary map was first introduced in [7]. It is clear that ∂R is a homomorphism of abelian groups. 6

Definition 4.2 ([13]) Let R be an HFD with quotient field K. We say that R is a boundary valuation domain (BVD) if given any α ∈ K ∗ with ∂R (α) 6= 0, either α ∈ R or α−1 ∈ R. It is easy to see that an HFD R with quotient field K is a BVD if for all α ∈ K ∗ with ∂R (α) > 0, α ∈ R (cf. Theorem 2.3 in [13]). Clearly, any rank-1 DVR is a BVD. A non-trivial class of examples includes domains of the form F + xK[[x]] where F ⊆ K is an extension of fields. Example 4.3 Let R be a BVD with complete integral closure R0 . Then R0 is a rank 1 DVR (cf. [13]). Denote the maximal ideal of R0 by zR0 . Then every irreducible of R is of the form uz for u ∈ U (R0 ). Of course, if x ∈ Irr(R), then G(x) is merely isomorphic to K1 . If ∂R (x) = 2, then we may write x = (u1 z)(u2 z) for ui ∈ U (R0 ). Let uz be any other irreducible of R nonassociate (in R) to u1 z and u2 z. Then u−1 u1 u2 z is an irreducible of R and clearly (uz)(u−1 u1 u2 z) = x. Therefore V (G(x)) = Irr(R). Now, suppose that (uz)(u1 z) ∈ E(G(x)). Then (up to a unit in R), (uz)(u1 z) = x = (u1 z)(u2 z) whence u = u2 , a contradiction. Therefore uz is not connected to either u1 z or u2 z in G(x). In fact, the only irreducible in R that is connected to uz is u−1 u1 u2 for if (uz)(vz) ∈ E(G(x)), then (again, up to a unit in R), (uz)(vz) = x = (u1 z)(u2 z) and v = u−1 u1 u2 . Therefore, in the case of ∂R (x) = 2, G(x) consists of the disjoint union of K2 s and single vertices with one loop (in the case where u1 u2 ∈ U (R)). Finally, suppose that ∂R (x) = n ≥ 3. Then we may write x = (u1 z)(u2 z) · · · (un z). By the same argument used above, V (G(x)) = Irr(R). So, let uz, vz ∈ Irr(R) be arbitrary. Then x = (u1 z)(u2 z) · · · (un z) = (uz)(vz)(u−1 v −1 u1 u2 u3 z) · · · (un z), and it follows that uz and vz are connected in E(G(x)), whence G(x) is a pseudo-clique on |V | vertices. Proposition 4.4 Let R be an atomic domain. Then the following are equivalent: 1. R is a BVD. 2. For all nonzero nonunit x in R, the following hold: a). Either x ∈ Irr(R) or V (G(x)) = Irr(R). b). If x can be written as a product of two irreducibles, then G(x) is a disjoint union of graphs, each of which is (graph) isomorphic to K2 or to a vertex with a single loop. c). If x can be written as the product of three or more irreducibles, then G(x) is a pseudo-clique on V (G(x)). 7

Proof: Example 4.3 shows that 1 implies 2. For the other implication, suppose that condition 2 holds. We will first show that R is an HFD. If x ∈ Irr(R), then we are done. Otherwise, suppose that bm x = π1a1 π2a2 · · · πnan = ξ1b1 ξ2b2 · · · ξm ,

with ai , bj ∈ N and πi , ξj ∈ Irr(R). We may assume that each of the π’s and ξ’s n P are pairwise nonassociate and that ai is the minimal length of any irreducible factorization of x. If

n P

i=1

ai = 2, then by property 2b, it is clear that

i=1 n P

Suppose next that

i=1

m P j=1

bj = 2.

ai = 3. Since G(x) is a pseudo-clique, ξ1 ∈ G( πx1 ).

However, πx1 can be written as a product of two irreducibles. It follows that m P bj = 3. j=1

n P

So we may now assume that 4 ≤

i=1

ai
0. Our aim is to show that α ∈ R. Let x = π1 · · · πn+t and let y = δ1 · · · δn . If n + t = 1 then n = 0 = ∂R (y), whence y ∈ U (R) and α ∈ R. 8

If n + t = 2 then either n = 0 (and y ∈ U (R)) or n = t = 1 and α =

π1 π2 . δ1

Since V (G(x)) = Irr(R), it follows that δ1 |x and α ∈ R. Suppose now that n + t ≥ 3. Again, since V (G(x)) = Irr(R), we have that δ1 |x. Thus there exists r ∈ R∗ such that δ1 r = x and ∂R (r) = ∂R (x) − 1. Also, r we have α = . Continuing inductively, we conclude that y|x and that δ2 · · · δn α ∈ R. ¥

5

A Characterization of Unique Factorization

We close with a characterization of unique factorization domains (or UFDs) via divisor graphs. Theorem 5.1 Let R be atomic. The following are equivalent: 1. R is a UFD. 2. For each nonzero nonunit x ∈ R, G(x) is a pseudo-clique. 3. For each nonzero nonunit x ∈ R, G(x) is a clique. 4. For each nonzero nonunit x ∈ R, G(x) is connected. Proof: It is clear that 2 ⇔ 3 and 2 ⇒ 4 both hold. (1 ⇒ 2,3): This assertion follows from Example 2.3. (2 ⇒ 1): Assume that R is not a UFD. Let A denote the set of all nonzero nonunit elements of R that do not factor uniquely into irreducibles. Assume that A is nonempty. We set n := min{m | z = δ1 δ2 · · · δm ; δi ∈ Irr(R)}. Choose x ∈ A and an z∈A

irreducible factorization of x, say x = π1a1 π2a2 · · · πnam with πi and πj nonassociate for each i 6= j and a1 + a2 + · · · + am = n. Since x ∈ A, we have another (distinct) irreducible factorization of x: bt , x = π1a1 π2a2 · · · πnam = ξ1b1 ξ2b2 · · · ξm

with ξi ∈ Irr(R), each ξi and ξj nonassociate for each i 6= j, and b1 + b2 + · · · + bt ≥ n. Note that no πi can be associate to any ξj , for if this were to occur, then we could cancel, violating the minimality of n. Now, we can factor πx1 as π1a1 −1 π2a2 · · · πnan . Since G(x) is a pseudo-clique, ξ1 is adjacent to π1 in G(x), and hence ξ1 is in the vertex set of G( πx1 ). This means that ξ1 is an irreducible divisor of πx1 . So we have the factorizations x = π1a1 −1 π2a2 · · · πnan = rξ1 π1 for some r ∈ R. Factoring r into irreducibles, we obtain x = π1a1 −1 π2a2 · · · πnan = α1 α2 · · · αt ξ1 . π1 9

Note that since ξ1 is pairwise nonassociate to any πi , the factorizations above are necessarily distinct. But this violates the minimality of a1 + a2 + · · · + am as the minimal length of factorization of any element of R admitting non-unique irreducible factorizations. Therefore A = {}, no such non-unique factorizations exist, and R is a UFD. (4 ⇒ 1): It will suffice to show that G(x) is a pseudo-clique for each x. Again, let A denote the set of all nonzero nonunit elements of R that do not factor uniquely into irreducibles. Assume that A is nonempty. We again set n := min{m | z = δ1 δ2 · · · δm ; δi ∈ Irr(R)}, and we take note that n ≥ 2. To z∈A

finish off the proof, we induct on n. First, suppose that n = 2. Let x ∈ A with x = yz, y, z ∈ Irr(R) nonassociate. Since x ∈ A, we have another factorization: x = yz = π1 π2 · · · πt , with πi ∈ Irr(R). Now G(x) is connected, so we may declare, without loss of generality that π1 is adjacent to y in G(x) and no πi is associate to z. To see why this is true, we first note that there is a path in G(x) between π1 and y, so we may replace π1 with the vertex on this path that is adjacent to y, implying that π1 is nonassociate to y. We may then take π1 π2 · · · πt to be an irreducible factorization of x involving π1 . If some πi , for 1 ≤ i ≤ t, were to be associate to some z, then we may cancel and conclude that t = 2 and that x factors uniquely, contradicting the fact that x ∈ A. Thus our claim is established. So, we have that π1 y|x, whence there exists r ∈ R∗ such that π1 yr = x = yz, implying that π1 is associate to z, a contradiction. If n = 2 and y is an associate to z (without loss of generality, y = z), then x = y 2 . The exact same argument above yields the same contradiction. Thus if x has a factorization consisting of exactly two irreducibles, then G(x) is a pseudo-clique. Next, we assume that n ≥ 3 and that any nonzero nonunit of R with minimal length factorization of length less than n factors uniquely. We now suppose that there exists x ∈ A with y1 y2 · · · yn a minimal length irreducible factorization of x. We will show that G(x) is a pseudo-clique. Since x ∈ A, we have another irreducible factorization x = y1 y2 · · · yn = π1 π2 · · · πt , where t ≥ n. Although the yi ’s may not be distinct, we claim that each yi is nonassociate to each πj . For if not, say without loss of generality, uy1 = π1 , where u ∈ U (R), then we would have u−1 y2 y3 · · · yn = π2 π3 · · · πt . However, (u−1 y2 )y3 · · · yn and π2 π3 · · · πt are nonassociate irreducible factorizations of yx1 (since y1 · · · yn and π1 · · · πt are nonassociate irreducible factorizations of x). We conclude that yx1 ∈ A, violating the minimality of n. Therefore no yi is associate to any πj . 10

The fact that G(x) is connected again implies that, without loss of generality, π1 and y1 are adjacent in G(x). Consider G( yx1 ). Since y1 π1 |x, π1 is a vertex in G( yx1 ), which is to say that π1 divides yx1 = y2 y3 · · · yn . By induction, G( yx1 ) is a pseudo-clique. Now, π1 , y2 , y3 , · · · , yn are all vertices in G( yx1 ). Since π1 is nonassociate to each of y2 , y3 , · · · , yn and since G( yx1 ) is a pseudo-clique, we must have π1 y2 | yx1 . Thus π1 ∈ G( y1xy2 ). x Continuing via use of induction, we see that π1 ∈ G( y1 y2 ···y ). However, n−1 x = y , implying that π and y are associate, a contradiction. Theren 1 n y1 y2 ···yn−1 fore A = {} and R is a UFD. ¥ Corollary 5.2 Let R be atomic and suppose that for all nonzero nonunit x ∈ R, G(x) is connected. Then each G(x) is a finite pseudo-clique. Proof: If G(x) is connected for all nonzero nonunit x ∈ R, then R is a UFD and hence an FFD. The result clearly follows. ¥ Corollary 5.3 Let R be atomic. Then R is a UFD if and only if for each nonzero nonunit x ∈ R, diam(G(x)) = 1. Proof: This follows from the fact that a graph has a diameter of 1 if and only if it is complete. ¥

References [1] S. Akbari, H. R. Maimani, and S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra 270 (2003), no. 1, 169–180. MR MR2016655 (2004h:13025) [2] S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra 274 (2004), no. 2, 847–855. MR MR2043378 (2004k:13009) [3] David F. Anderson, Andrea Frazier, Aaron Lauve, and Philip S. Livingston, The zero-divisor graph of a commutative ring. II, Ideal theoretic methods in commutative algebra (Columbia, MO, 1999), Lecture Notes in Pure and Appl. Math., vol. 220, Dekker, New York, 2001, pp. 61–72. MR MR1836591 (2002e:13016) [4] David F. Anderson, Ron Levy, and Jay Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra 180 (2003), no. 3, 221–241. MR MR1966657 (2003m:13007) [5] David F. Anderson and Philip S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434–447. MR MR1700509 (2000e:13007)

11

[6] Istv´an Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226. MR MR944156 (89i:13006) [7] J. Coykendall, The half-factorial property in integral extensions, Comm. Algebra 27 (1999), no. 7, 3153–3159. MR 2000d:13013 [8] F. R. DeMeyer, T. McKenzie, and K. Schneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum 65 (2002), no. 2, 206–214. MR MR1911724 (2003d:20091) [9] Frank DeMeyer and Lisa DeMeyer, Zero divisor graphs of semigroups, J. Algebra 283 (2005), no. 1, 190–198. MR MR2102078 (2005h:20138) [10] Frank DeMeyer and Kim Schneider, Automorphisms and zero divisor graphs of commutative rings, Commutative rings, Nova Sci. Publ., Hauppauge, NY, 2002, pp. 25–37. MR MR2037656 (2004k:13010) [11] Ron Levy and Jay Shapiro, The zero-divisor graph of von Neumann regular rings, Comm. Algebra 30 (2002), no. 2, 745–750. MR MR1883021 (2002k:13014) [12] Dan-cheng Lu and Wen-ting Tong, The zero-divisor graphs of abelian regular rings, Northeast. Math. J. 20 (2004), no. 3, 339–348. MR MR2081903 (2005e:16017) [13] J. Maney, Boundary valuation domains, J. Algebra 273 (2004), 373–383. [14] Shane P. Redmond, The zero-divisor graph of a non-commutative ring, Commutative rings, Nova Sci. Publ., Hauppauge, NY, 2002, pp. 39–47. MR MR2037657 [15]

, An ideal-based zero-divisor graph of a commutative ring, Comm. Algebra 31 (2003), no. 9, 4425–4443. MR MR1995544 (2004c:13041)

[16]

, Structure in the zero-divisor graph of a noncommutative ring, Houston J. Math. 30 (2004), no. 2, 345–355 (electronic). MR MR2084907 (2005d:16002)

[17] Tongsuo Wu, On directed zero-divisor graphs of finite rings, Discrete Math. 296 (2005), no. 1, 73–86. MR MR2148482

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