Abstract

An elementary annihilator of a ring is an annihilator that has the form ; . We define the elementary annihilator dimension of the ring , denoted by , to be the upper bound of the set of all integers such that there is a chain of annihilators of . We use this dimension to characterize some zero-divisors graphs.

1. Introduction

In this paper, all rings are considered to be commutative and unitary.

Let be a ring and be a nonempty subset of . We call the annihilator of in denoted by or the set . If is a singleton then will be denoted by . If then is called an elementary annihilator. An annihilator is said to be maximal if it is maximal in the set of all proper annihilators of . It is well known that all maximal annihilators are elementary. For an elementary annihilator chain is said to be a chain of elementary annihilators with length ending in . The upper bound of the set of all lengths of elementary annihilator chains ending in is called the elementary annihilator height of (or ). In this paper, we introduce a dimension of a ring using elementary annihilator chains called elementary annihilator dimension, denoted by . The is the upper bound of the set of elementary annihilator heights. We use this dimension to study zero-divisor graphs.

We introduce a class of rings called isometric maximal elementary annihilator rings, in short IMEA-rings. That is the class of rings with finite EAdimension whose all maximal annihilators have the same height.

2. Elementary Annihilator Dimension of a Ring

Definition 1. (1) Let and be chain of elementary annihilators in the ring . One says that this chain is an elementary annihilator chain of length ending in .
(2) Let be a nonzero element of . One defines the elementary annihilator height of , denoted by , as the upper bound of the set of all lengths of elementary annihilator chains ending in .
(3) One calls elementary annihilator dimension of , denoted by , the upper bound of the set .

Example 2. (1) . Indeed, is the longest chain of elementary annihilators in .
(2) .
(3) All nonzero zero-divisors satisfy . Indeed, is a chain of length one.

It is easy to check the following results.

Remark 3. (1) Let , if and only if is regular.
(2) if and only if is a domain.
(3) For an ideal of , if and only if is prime.
(4) If is a nonzero noninvertible element if and only if is prime.

We denote by the set of all nilpotent elements of . is said to be reduced if it has no nilpotents other then zero.

Theorem 4. Let and be its index of nilpotency; one has:   .

Proof. If or is infinite the result is obvious. Otherwise, there exists a chain whose length is and it ends in . Let be this chain. Moreover, we have . So we obtain the following chain:     whose length is . Consequently, .

Corollary 5. If satisfies is finite then . In particular, if then for all , .

Theorem 6. Let and be two rings; then;(1) is finite if and only if and are finite.(2).

Proof. Let be a nonzero zero-divisor. If and are nonzero then .
If one of them is zero, for example, then .
(1) “” Let , suppose that or is infinite; for example, is infinite. Then there exists and a chain in ; then is a chain of elementary annihilators in whose length is , contradiction.
“” If we assume that , then there is a chain of length in ; let be this chain. In the same way we put and we take as a chain of length . Then    ,   is an elementary annihilator chain of whose length is that is maximal, because of the inclusion or . Then is finite and .
(2) If is infinite (that is or is infinite, by (1)) then the result is obvious. The finite case is shown in the proof of (1) “”.

By induction, we have the following result.

Corollary 7. (1) Let be some rings, one has .
(2) If is a domain and then .

3. The EAdimension and the Zero-Divisor Graph

Let be a ring. The zero-divisor graph of is defined to be the graph whose vertices are the nonzero zero-divisors of and its edges are the pairs satisfying . We denote this graph by . For the simplicity of writing we still denote by the set of nonzero zero-divisors of .

is said to be connected if for every two different vertices and of there is a sequence such as , and is an edge, . This sequence is called a path connecting and with length . is said to be complete if each two distinct vertices form an edge. We call the distance between and the least length of a path connecting them, denoted by or . We call the diameter of , denoted , the supremum of the set . In [1], Anderson and Livingston showed that is connected and .

For an integer , Anderson and Livingston defined to be -partite complete if , where the s are nonempty disjoined sets and for all in satisfy if and only if there exists such that . In this paper we extend the definition of r-partite complete graph to the case when is infinite.

Lemma 8 (see [2], Theorem 6). (1) If is an elementary annihilator that is maximal (in the set of proper annihilators of ) then it is prime.
(2) Let ; if is maximal and then .

Proposition 9. Let be a reduced ring that is not a domain and be two nonzero zero-divisors such that and are maximal; then if and only if .

Proof. “” Immediately, by Lemma 8 “” If , suppose that .   , contradiction. Then .

Theorem 10. If is a nonreduced ring then if and only if is complete.

Proof. “” if is complete then, according to [1, Theorem 2.8], we have for all , . Then , . So all nonzero elementary annihilators are equal, and then .
“” Let ; then , according to Corollary 5. If then is complete. Otherwise, for all we have and are maximal. Suppose that , according to Lemma 8(2), and there, for example, ; then , contradiction to the maximality of . Then and . Consequently, and all nonzero zero-divisors satisfy . It follows that for all , . According to [1, Theorem 2.8], is complete.

Theorem 11. If is partite complete graph with , then under one of the following conditions: (1)two of the 's contain, each one, more than one element;(2) is reduced.

Proof. Suppose that .
First case: If two among the 's contain, each one, more then one element. Assume that , where and have, each one, at least two elements. Let , and , . We have ; then is a divisor of zero in . Suppose that ; then , contradiction. Then . Suppose that ; then , contradiction; then . If then , contradiction. Then and   , contradiction.
Second Case: If Is Reduced. Assume that . Let , , we have then is a divisor of zero in . Suppose that , then , contradiction. Then . Suppose that then  , contradiction. Then . If   then  , contradiction. Then   and then   , contradiction. We conclude that .

Theorem 12. If is reduced then if and only if is bipartite complete.

Proof. “” In , we define the relation “” by the following: if . is a relation of equivalence. For all , we denote by its equivalence class. The different classes form a partition of and we write . Since is reduced and not a domain, then there exist nonzero elements satisfying . Now, ; then and are maximal. According to Proposition 9, ; then ; then .
If then then , by Proposition 9.
Let ; then ; then, by Lemma 8, . And we conclude that is -partite complete. According to Theorem 11, is bipartite complete.
“” Assume that is bipartite complete. If then , for . Let , if then . Otherwise and are incomparable. Then for all , . It follows that .

Theorem 13. Let be a ring.(1)If then .(2)If or and is reduced then .(3)If and is not reduced or then .

Proof. (1) If is reduced then, by Theorem 12, is bipartite complete; then .
If is not reduced then, by Theorem 10, is complete then, by Theorem 2.8 of [1], for all , . Then, by Theorem 2.6 of [3], .
(2) If and is reduced then, by Theorem 2.6 of [3], is reduced with exactly two minimal primes and at least three nonzero zero-divisors. Then , where , are the two minimal primes of ; they satisfy . Then for all and , and for , . Consequently, is bipartite complete graph. According to Theorem 12, .
If : if , by Theorem 2.6 of [3], is isomorphic to either or and in both cases has a unique nonzero elementary annihilator then .
Now, if , using Theorem 2.6 of [3], for each distinct pair of zero-divisors and has at least two nonzero zero-divisors. According to Theorem 2.8 of [1], is complete and is not reduced. By Theorem 10, .
(3) If then, by (1), . Since then is not a domain then . Consequently, . If and is not reduced: then is not a domain then . Suppose that ; then by Theorem 10, is complete then , contradiction. Then .

Lemma 14. Let be a ring, and , a set of distinct prime ideals which are incomparable. For an element of we denote . Let .(1)For all subsets of there is such that .(2)Let , . In particular, .(3)Let and be two nonempty subsets of . If then .(4)Let . .(5)The map is a decreasing bijection, here denotes the set of subsets.

Proof. (1) If , take then , for all then and the result is true in this case.
If , take then , for all then .
Now if : , and , . Suppose that ; then there exists such that ; then there exists such that . Since the ’s in are incomparable under inclusion then , then , contradiction. Consequently, there exists such that .
(2) Let ,     and and , and , and . In the last equivalence the indirect sense “” is obtained by (1).
(3) Suppose that and , where the ’s (resp., ’s) are pairwise different.    then one of the 's is contained in ; for example, . Since the elements of are incomparable under inclusion then .
   then one of the ’s is contained in . Suppose that ; then then , contradiction. Then there exists such that ; for example, ; then .
We repeat this process until reaching the stage number .
Suppose that ; then ; then there exists such that that is, , contradiction. Consequently, and we get , ; then .
(4) Let , .
(5) We check that is well defined: put . According to (4), . Then ; then . According to (3), .
For ,  ; then . Then is well defined.
is injective, by (4).
We show that is surjective: let . According to (1), there exists such that . Since then and .
. Then, according to (3), . Thus is a decreasing bijection.