On the Structures of Abelian -Regular Rings

Assume that is an Abelian ring. In this paper, we characterize the structure of whenever is -regular. It is also proved that an Abelian -regular ring is isomorphic to the subdirect sum of some metadivision rings. 1. Introduction Throughout this paper, we only consider the associative rings with identity . By , we denote the full set of idempotent elements of the ring . Write to denote the center of the ring , and write , the full set of nilpotent elements of . The ring is called Abelian if . The ring is said to be regular if, for any , there exists an such that . And is called -regular if, for any , there exist and a positive integer such that . We also say a ring is strongly -regular if, for any , there are and positive integer such that . It follows that strongly -regular rings are -regular and Abelian -regular rings are strongly -regular. The ring is said to be a metadivision if any element in is either a unit or a nilpotent element. In ring theory, there has been much interest in finding connections between the -regularity and the condition that every prime ideal is maximal, which has been studied, for example, in [1–6]. A pretty result due to Storrer in [6] is if is a commutative ring with identity then is -regular if and only if every prime ideal of is maximal. A commutative ring is obviously an Abelian ring; thus, the following result is a generalization of Storrer’s result. Theorem A. Let be an Abelian ring. Then is -regular if and only if the following statements hold:(1) is an ideal of ;(2)every one-sided ideal containing is an ideal in ;(3)every completely prime ideal is a maximal ideal in . A classical result due to McCoy in [7] states that a commutative -regular ring is isomorphic to the subdirect sum of some metadivision rings. We generalize this result as follows. Theorem B. Assume that is an Abelian -regular ring and denotes the collection of all completely prime ideals of . Then is isomorphic to the subdirect sum of the ’s for all . 2. Abelian -Regular Rings Recall that an ideal of is primary if for any ; we always obtain or for some positive integer . Definition 1. The ring is said to be a metadivision ring if any element in is either a unit or a nilpotent element. In fact, there is an intimate connection between the primary ideas and the metadivision rings as follows. Lemma 2. Let be an Abelian -regular ring. Then is a primary ideal of if and only if is a metadivision ring. Proof. The “only if” part: for any element , there exists the equality for a positive integer and . Set ; then is an idempotent. If is not nilpotent, then and so ;