The EA-Dimension of a Commutative Ring

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