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Volume 2013 (2013), Article ID 293207, 6 pages
The EA-Dimension of a Commutative Ring
Institut Préparatoire aux Études D’ingénieurs de Monastir, Tunisia
Received 29 October 2013; Accepted 21 November 2013
Academic Editors: D. Anderson and A. V. Kelarev
Copyright © 2013 Mosbah Eljeri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 .
(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.
(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 , 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 , 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 , for all , . Then, by Theorem 2.6 of , .
(2) If and is reduced then, by Theorem 2.6 of , 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 , is isomorphic to either or and in both cases has a unique nonzero elementary annihilator then .
Now, if , using Theorem 2.6 of , for each distinct pair of zero-divisors and has at least two nonzero zero-divisors. According to Theorem 2.8 of , 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.
Theorem 15. Let be a ring and be different incomparable prime ideals of . Then .
Proof . If then is a domain and the result is checked.
Now let . is decreasing sequence in with maximal length. Using the bijection defined in Lemma 14, is a chain of elementary annihilators in that has a maximal length. Then .
Example 16. A ring is said to be semilocal ring if it has a finite number of maximal ideals. Let be a semilocal ring with maximal ideal; then , where is the Jacobson radical of . We obtain this result by using the previous theorem.
A ring is called a noetherian spectrum ring if it satisfies the ascending chain condition (acc) on radical ideals; equivalently each radical ideal is a radical of finitely generated ideal. The set of prime ideals of a ring which are minimal over an ideal , denoted by , is finite in the case when is a noetherian spectrum ring. If , we denote by instead of . For more information about noetherian spectrum rings see [4, Chapter 2].
Proposition 17. Let be a noetherian spectrum ring, for all ideals , .
Definition 18 (according to ). Let be a ring.(1)One calls the chromatic number of the minimal number of colors used to color the elements of such that each two adjacent elements (with zero product) have different colors, denoted by .(2)One says that the ring is a coloring if its chromatic number is finite.
Theorem 19. If is a reduced coloring then .
Proof. If is a reduced coloring, according to [5, Theorem 3.8], is finite. And if then . Let ; then . According to Theorem 15, .
Let be a ring such that . For all vertices in , we denote and . For all , we denote by the subgraph of whose vertices form the following set et .
Theoretically we can write .
Remark 20. Let be a ring such that . If then . Indeed, consider and . There exists and such that . Suppose that then . According to Lemma 8(2) , this contradicts the fact that . Then and is an edge. Now take , three cases are possible. If then take and the chain is of length 2 then . If only is in then is an edge. If then take and is a chain of length 2. Then, in all cases, that is . Now , then there exists such that . Let then then . Consequently, .
4. Isometric Maximal Elementary Annihilator Rings
Definition 21. Let be a ring with finite EA dimension; one says that is an isometric maximal elementary annihilator ring, in short an IMEA-ring if its all maximal elementary annihilators have the same height.
Example 22. The ring is an IMEA-ring. Indeed, the elementary annihilators of are , , and . They satisfy and .
Theorem 23. Let and be two rings; then is an IMEA-ring if and only if and are two IMEA-rings.
Proof. By Theorem 6(1), is finite if and only if and are finite.
is a maximal elementary annihilator in if and only if , and is a maximal elementary annihilator in or , and is a maximal elementary annihilator in , then all maximal elementary annihilators of have the same hight if and only if all maximal elementary annihilators of have the same height and also for the maximal elementary annihilators of .
We get the following result, inductively.
Corollary 24. Let be some rings. We have is an IMEA-ring if and only if each is an IMEA-ring.
Let be a domain, we say that is atomic if each nonzero nonunit element of decomposes into a finite product of irreducibles, according to . An atomic domain is called a half factorial domain, in short a HFD if are two decompositions into irreducibles then . This concept was introduced by Zaks in . A HFD is called a unique factorization domain, in short a UFD if are two decompositions into irreducibles then the ’s and the ’s are associates after reordering them. It is well known that a UFD is an atomic domain in which each irreducible is primed by [8, Theorem 1].
Proposition 25. If is a UFD, then for all nonzero nonunit of we have is an IMEA-ring. Moreover if is the decomposition of into prime elements then .
Proof. Let . Suppose that ; then
And for (one among the ’s is and ), we have the following:
It is easy to check that the set of all elementary annihilators of is The maximal ones among them are A longest chain ending in one of them, for example, , has the length . Thus all maximal elementary annihilators have the same height then is an IMEA-ring and .
Proposition 26. Let be a HFD. is a UFD if and only if , for all nonzero nonunit . Where is the number of factors in a decomposition of into irreducibles (counted with multiplicities).
Question 1. Are all finite EAdimensional rings IMEA-rings?
Question 2. Are all finite rings IMEA-rings?
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