GENERALIZATIONS OF THE ZERO-DIVISOR GRAPH

Let R be a commutative ring with 1 6= 0 and Z(R) its set of zerodivisors. The zero-divisor graph of R is the (simple) graph Γ(R) with vertices Z(R) \ {0}, and distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we consider generalizations of Γ(R) by modifying the vertices or adjacency relations of Γ(R). In particular, we study the extended zero-divisor graph Γ(R), the annihilator graph AG(R), and their analogs for ideal-based and congruence-based graphs. Mathematics Subject Classification (2010): 05C99, 13A15, 13A99


Introduction
Let R be a commutative ring with 1 = 0, and let Z(R) be its set of zero-divisors.
The zero-divisor graph of R is the (simple) graph Γ(R) with vertices Z(R) * = Z(R)\ {0}, the set of nonzero zero-divisors of R, and distinct vertices x and y are adjacent if and only if xy = 0. There have been several other "zero-divisor" graphs associated to R. The extended zero-divisor graph of R is the (simple) graph Γ(R) with vertices Z(R) * , and distinct vertices x and y are adjacent if and only if x m y n = 0 for positive integers m and n with x m = 0 and y n = 0. The annihilator graph of R is the (simple) graph AG(R) with vertices Z(R) * , and distinct vertices x and y are adjacent if and only if ann R (xy) = ann R (x) ∪ ann R (y) (i.e., ann R (x) ∪ ann R (y) ann R (xy)).
by F. R. DeMeyer, T. McKenzie, and K. Schneider in [22], and the semigroup annihilator graph AG(S) was defined by M. Afkhami, K. Khashyarmanesh, and S.
Starting with Γ(R), we can modify both the vertices and edges to get new "zerodivisor" graphs. In this paper, we study these ideas in more detail. In the second section, we study the ideal-based graphs Γ I (R), Γ I (R), and AG I (R). Results for Γ(R), Γ(R), and AG(R) then follow by letting I = {0}. In the third section, we study the congruence-based graphs Γ ∼ (R), Γ ∼ (R), and AG ∼ (R). We also investigate when Γ ∼ (R) is complemented or uniquely complemented. In the fourth section, we study compressed graphs. In the final section, we study maps between congruence-based graphs. This extends the work in [12] on Γ ∼ (R) to Γ ∼ (R) and AG ∼ (R).
Throughout, R will be a commutative ring with 1 = 0, Z(R) its set of zerodivisors and Z(R) * = Z(R) \ {0}, nil(R) its set of nilpotent elements, U (R) its group of units, T (R) = R S , where S = R \ Z(R), its total quotient ring, and dim(R) its Krull dimension. For I an ideal of R, √ I = { x ∈ R | x n ∈ I for some n ∈ N }, and I is a radical ideal if √ I = I. For I an ideal of R and x ∈ R, let (I : x) = { y ∈ R | xy ∈ I }; thus (0 : x) = ann R (x). We say that R is reduced if nil(R) = {0}. These concepts extend in the obvious way to semigroups and semigroup ideals. We will often consider R to be a commutative monoid under the ring multiplication. In this case, an (ring) ideal of R is always a semigroup ideal of R. However, the converse may fail since the union of (ring) ideals of R is a semigroup ideal of R. As usual, we assume that a subring has the same identity element as the ring R, x 0 = 1 for 0 = x ∈ R, and all ring and monoid homomorphisms send 240 DAVID F. ANDERSON AND GRACE MCCLURKIN the identity to the identity. Let N, Z, Z n , Q, and F q denote the positive integers, integers, integers modulo n, rational numbers, and the finite field with q elements, respectively, and A * = A \ {0}. For any undefined ring-theoretic terminology, see [26] or [27]; for semigroups, see [25].
Except for a brief digression in Section 4, we assume that all graphs are simple graphs, i.e., they are undirected graphs with no multiple edges or loops. By abuse of notation, we will often let G, rather than V (G), denote the vertices of a graph As usual, K n will denote the complete graph on n vertices, and K m,n will denote the complete bipartite graph on m, n vertices (m and n may be infinite cardinals).
If m = 1 (or n = 1), then K m,n is called a star graph. A subgraph H of a graph only if f (x) and f (y) are adjacent in G ); in this case, we write G ∼ = G (again, by abuse of notation, we will often just write G = G when f is a naturally induced when the context is clear.) To avoid trivialities, we will implicitly assume when necessary that our graphs are not the empty graph. A general reference for graph theory is [20].
Many of the results in this paper are from the second-named author's PhD dissertation ( [28]) at The University of Tennessee under the direction of the firstnamed author.

Ideal-based graphs
In this section, we study the ideal-based graphs Γ I (R), Γ I (R), and AG I (R).
These graphs all have common vertex set Z I (R), but different adjacency relations. We will prove results for the general ideal-based case, and then results for There is a strong relationship between Γ I (R) and Γ(R/I). In fact, Γ I (R) may be constructed from Γ(R/I) and I; see [31, Section 2] for details. A similar relationship and construction exist for Γ I (R) and AG I (R)) ( [28]).
It is clear that Γ I (R) is a subgraph of both Γ I (R) and AG I (R). We first show that Γ I (R) is also a subgraph of AG I (R).
Theorem 2.1. Let R be a commutative ring with 1 = 0 and I an ideal of R. Then Proof. Clearly Γ I (R) ⊆ Γ I (R). We show that Γ I (R) ⊆ AG I (R). Let x and y be adjacent vertices in Γ I (R). Then x m y n ∈ I for positive integers m and n with x m ∈ I and y n ∈ I. We may assume that x m−1 y n ∈ I and x m y n−1 ∈ I. Let z = x m−1 y n−1 . Then z(xy) = x m y n ∈ I, but zx = x m y n−1 ∈ I and zy = x m−1 y n ∈ I; so z ∈ (I : xy) \ ((I : x) ∪ (I : y)). Thus x and y are also adjacent in AG I (R); so The following example shows that all possible inclusions may occur. It is easy to check that (1) Γ(R) = Γ(R) = AG(R) = K 1 for R = Z 4 , (2) Remark 2.4. One is tempted to define the extended annihilator graph AG(R) with vertices Z(R) * , and distinct vertices x and y are adjacent if and only if ann R (x m y n ) = ann R (x m ) ∪ ann R (y n ) for positive integers m and n. However, For the reverse inclusion, let x and y be adjacent vertices in AG(R). Then ann R (x m y n ) = ann R (x m ) ∪ ann R (y n ) for positive integers m and n, and thus zx m y n = 0, but zx m = 0 and zy n = 0, for some z ∈ R. We may assume that zx m y n−1 = 0 and zx m−1 y n = 0. Thus zx m−1 y n−1 ∈ ann R (xy) \ (ann R (x) ∪ ann R (y)), and hence x and y are also adjacent in AG(R). Thus AG(R) = AG(R). In a similar manner, we can define AG I (R), and as above, AG I (R) = AG I (R).
Although distinct, nonzero, nilpotent elements x and y need not be adjacent in Γ(R), we have d(x, y) ≤ 2 in Γ(R), and they are always adjacent in AG(R). We generalize this to Γ I (R) and AG I (R).
Theorem 2.5. Let R be a commutative ring with 1 = 0, I an ideal of R, and x, y ∈ √ I \ I with x = y.
(b) x and y are adjacent in AG I (R).
Proof. (a) Let x, y ∈ √ I \ I be distinct with xy ∈ I. Let m and n be the least positive integers such that x m , y n ∈ I. Choose j to be the greatest nonnegative integer such that z = x m−1 y j ∈ I. Then xz = x m y j ∈ I and zy = x m−1 y j+1 ∈ I by choice of j; so d(x, y) = 2 in Γ I (R).
(b) Let x, y ∈ √ I \ I be distinct. Let m and n be the least positive integers such that x m , y n ∈ I; so x m y n ∈ I. If x i y j ∈ I for some 1 ≤ i ≤ m − 1, 1 ≤ j ≤ n − 1, then x and y are adjacent in Γ I (R), and hence in AG I (R) by Theorem 2.1. So we may assume that x i y j ∈ I for every 1 ≤ i ≤ m − 1 and 1 ≤ j ≤ n − 1. In Then zxy = x m y n−1 + x m−1 y n ∈ I, and zx = x m y n−2 + x m−1 y n−1 ∈ I since  (resp., [30, Theorem 3.1]).) As in [16], an ideal I of R is a 2-absorbing ideal of R if whenever xyz ∈ I for x, y, z ∈ R, then xy ∈ I, xz ∈ I, or yz ∈ I.  (1) x ∈ √ I implies x 2 ∈ I, and (2) x ∈ R \ √ I implies (I : x 2 ) = (I : x).
In particular, Γ I (R) = Γ I (R) when I is a radical ideal of R.
Proof. (a) Suppose that Γ I (R) = Γ I (R). First, let x ∈ √ I. If x n ∈ I, but x n−1 ∈ I for n ≥ 3, let y = x(1 + x n−2 ). Then y ∈ √ I \ I (y ∈ I since y ∈ I implies x 2 = xy − x n ∈ I), x = y, and x and y are adjacent in Γ I (R) since x n−1 y = x n (1 + x n−2 ) ∈ I. Thus x and y are also adjacent in Γ I (R); so xy ∈ I.
Hence x 2 = xy − x n ∈ I, a contradiction; so x 2 ∈ I and (1) holds. Next, let x ∈ R \ √ I and y ∈ (I : x 2 ). If y ∈ I, then y ∈ (I : x). If x = y, then x 3 ∈ I, a contradiction. So assume that y ∈ I and x = y. Then x 2 y ∈ I; so x and y are adjacent in Γ I (R). Hence they are also adjacent in Γ I (R); so xy ∈ I. Thus (I : x 2 ) ⊆ (I : x); so (2) holds.
Conversely, suppose that (1) and (2)  Similarly, xy ∈ I if x ∈ √ I. So in every case, xy ∈ I, and hence x and y are also adjacent in Γ I (R). Thus Γ I (R) = Γ I (R).
The "in particular" statement is clear.
(b) First, suppose that AG I (R) = Γ I (R). Let xyz ∈ I for x, y, z ∈ R. There are three cases.
(1) Suppose that x = y = z; then x 3 ∈ I. If x 2 ∈ I, then x and w = x(1+x) are distinct adjacent vertices in AG I (R), and hence are also adjacent in Γ I (R).
(2) Suppose that x = y and z are distinct; then x 2 z ∈ I. Suppose that x 2 ∈ I.
If xz ∈ I, then x ∈ (I : xz) \ ((I : x) ∪ (I : z)). Thus x and z are adjacent vertices in AG I (R), and hence are also adjacent in Γ I (R). Thus xz ∈ I, a contradiction. So xz ∈ I.
(3) Suppose that x, y, z are all distinct, and xz ∈ I and yz ∈ I. Then z ∈ (I : xy) \ ((I : x) ∪ (I : y)); so x and y are adjacent in AG I (R). Thus x and y are also adjacent in Γ I (R); so xy ∈ I.
Thus I is a 2-absorbing ideal of R.
Conversely, suppose that I is a 2-absorbing ideal of R.
Theorem 2.10. Let R be a commutative ring with 1 = 0 and total quotient ring , and let ∼ R (resp., . These results also hold for the ideal-based graphs and have been investigated in [31] and [15] for Γ I (R) and in [1] for AG I (R).
Theorem 2.11. Let R be a commutative ring with 1 = 0 and I an ideal R.
Proof. . So we may assume that I is a nonzero, proper ideal of R that is not The "moreover" statement is clear since Corollary 2.12. Let R be a commutative ring with 1 = 0.
The following relationship between diam(Γ(R)) and diam(AG(R)) will be needed in Theorem 2.14.
Theorem 2.13. Let R be a commutative ring with 1 = 0. (b) Suppose that AG(R) is complete. We may assume that |Z(R) * | ≥ 3; so is an ideal of R by part (a). Let x, y ∈ Z(R) * be distinct.
Then x − y ∈ Z(R); so there is a 0 = z ∈ R with z(x − y) = 0. We may assume that z = x. Since AG(R) is complete, z and x are adjacent in AG(R). Hence The "in particular" statement is clear.
We next give a more careful comparison of the diameter and girth for the three zero-divisor graphs.

Congruence-based graphs
In this section, we consider congruence-based graphs. For a more detailed account of the ∼-zero-divisor graph Γ ∼ (R), see [12]. In Section 5, we consider maps between congruence-based graphs.
We first define the congruence-based analogs of Γ(R) and AG(R). Let ∼ be a multiplicative congruence relation on R. Then R/∼ is a commutative monoid (2) x ∈ R \ √ I implies (I : x 2 ) = (I : x).
In particular, Γ ∼ (R) = Γ(R) ∼ when I is a radical semigroup ideal of R.

DAVID F. ANDERSON AND GRACE MCCLURKIN
First, suppose that Γ ∼ (R) Γ ∼ (R). By Theorem 3.1(b), either (1) there is an x ∈ R such that x n ∈ I, but x n−1 ∈ I for some n ≥ 3, or (2) there is an x ∈ R \ √ I such that (I : x) (I : x 2 ). If (1) holds and n ≥ 4, then is a 3-cycle in Γ ∼ (R). So suppose that n = 3. The "moreover" statement is clear.
Let G be a simple graph. We say that distinct adjacent vertices x and y of G are orthogonal, written x ⊥ y, if there is no vertex z of G adjacent to both x and y (i.e., the edge x -y is not part of a triangle in G). The graph G is . We extend these results to Γ ∼ (R).
We will need the following lemma. Note that [x] ∼ = [y] ∼ does not imply that x + I = y + I, or conversely. The "moreover" statement is clear. (1) Γ ∼ (R) is uniquely complemented.
The compressed zero-divisor graph Γ E (R) "compresses" Γ(R) by identifying vertices with the same adjacency relations. However, Γ E (R) and AG E (R) need not "compress" Γ(R) and AG(R) in this way (see Example 4.2). We next define the "compressed" graphs associated to Γ(R) and AG(R). Although all the graphs we have studied in this paper are simple graphs, it is sometimes convenient to consider these graphs to have loops. For example, there is a loop at x ∈ Z(R) * in Γ(R) (resp., Γ(R), AG(R)) if and only if x 2 = 0 (resp., x ∈ nil(R), ann R (x 2 ) = ann R (x)). We will denote the graph Γ(R) (resp., Γ(R), AG(R)) with loops added by Γ L (R) (resp., Let G be Γ(R), Γ(R), or AG(R). For x ∈ Z(R) * , define their extended edge sets . For the graphs G = Γ(R), Γ(R), and AG(R), the relation ∼ G defined by The equivalence relation ∼ G on Z(R) * may be extended to an equivalence relation on R by defining [0] ∼ G = {0} and [1] ∼ G = R \ Z(R). We will see that ∼ Γ(R) is actually a multiplicative congruence relation on R. However, the following example shows that ∼ G need not be a multiplicative congruence relation on R (or Z(R) * ) when G is Γ(R) or AG(R).
We next consider some cases when CΓ(R) = Γ E (R) and CAG(R) = AG E (R).
Although the next result may seem obvious, note that we need to consider loops.
for every x ∈ Z(R) * since CAG(R) and AG E (R) have the same adjacency relations. Since for every x ∈ Z(R) * . So we need only show that x 2 = 0 ⇔ ann R (x 2 ) = ann R (x). The (⇒) implication is clear; so suppose by way of contradiction that ann R (x 2 ) = ann R (x), but x 2 = 0. Let y ∈ ann R (x 2 ) \ ann R (x); so yx 2 = 0 and yx = 0. If y = x, then implies ann R (x 2 ) = ann R (x). For (1), assume by way of contradiction that x n = 0, but x n−1 = 0, for some integer n ≥ 3.

Maps between graphs
In this section, we study maps between congruence-based graphs. This extends the work in [12] on Γ ∼ (R) to Γ ∼ (R) and AG ∼ (R).
First, we recall several results from [12]. For a commutative ring R, let C(R) be the set of multiplicative congruence relations on R. We can partially order C(R) by ∼ 1 ≤∼ 2 ⇔ ∼ 1 ⊆∼ 2 (as subsets of R × R). So C(R) has a least element Let I be a semigroup ideal of R and ∼ 1 , Since F is surjective, there is an (not necessarily unique) injective function choose an α(z) ∈ Z(R/∼ 1 ) * such that F (α(z)) = z, and then define G(z) = α(z)). We need only check adjacency. Note that the functions F, F , F (resp., G, G , G ) take the same values on Z(R/∼ 1 ) * (resp., Z(R/∼ 2 ) * ), but to avoid any possible confusion on which graphs are being considered, we will use the " s".
Theorem 5.3. Let R be a commutative ring with 1 = 0, I a radical semigroup ideal of R, and ∼ 1 , ∼ 2 ∈ C I (R) with ∼ 1 ≤ ∼ 2 . Then F : Γ ∼1 (R) −→ Γ ∼2 (R) (resp., Most of the results in [12, Sections 5 and 6] for Γ ∼ (R) extend in the natural way to Γ ∼ (R) and AG ∼ (R) since all three graphs have the same vertex set. We leave the routine details to the interested reader.