AL-Rafidain Journal of Computer Sciences and Mathematics

Let R be a commutative ring with identity and AG(R) be the set of ideals with      non-zero annihilators. The annihilating ideal graph AG(R) is a graph of vertex set             AG (R)\{(0)}  and two distinct ideal vertices I  and J are adjacent if and only if  . In this paper , we establish a new fundamental properties of AG(R) as well as its connection with Γ(R).
 

As a generalization of regular rings, we introduce the notion, of m-regular rings, that is for all , there is a fixed positive integer m such that  is a Von-Neumann regular element. Some characterization and basic properties of these rings will be given. Also, we study the relation-ship between them and Von-Neumann regular rings, -regular rings, reduced rings, locally rings, uniform rings and 2-primal rings.
 

As a generalization of right weakly regular rings, we introduce the notion of right n-weakly regular rings, i.e. for all aN(R), aaRaR. In this paper, first  give  various properties of right n-weakly regular rings. Also, we study the relation between such rings and reduced rings by adding some types of rings, such as NCI, MC2 and SNF rings.
 

As a generalization of regular rings, that is a ring is called n-regular if  for all . In this paper, we first  give  various properties of n-regular rings. Also, we study the relation between such rings and reduced rings by adding some types of rings, such as NCI rings, and other types of rings.
 

A right module M is called Wnil-injective if for any 0 ¹ a Î N( R ), there exists a positive integer n such that aⁿ ¹ 0 and any right R-homomorphism f : aⁿR ® M can be extended to R ® M. In this paper , we first give and develope various properties of right Wnil-injective rings, by which, many of the known results are extended. Also, we study the relations between such rings and reduced rings by adding some types of rings, such as SRB-rings, and other types of rings.
 

A ring R is said to be right(left) sp-weakly regular if for each a Î R and a positive integer n,  aΠ aR aR (aÎ R aR a). In this paper, we continue to study sp-weakly regular rings due to R. D. Mahmood and A. M. Abdul-Jabbar [8]. We first consider properties and basic extensions of sp-weakly regular rings, and we give the connection of sp-weakly regular, semi  p-regular and p-biregular rings.
 
 

The main purpose of this paper is to study CS-rings. We give some properties of right CS-rings and the connection between such rings and reduced rings, regular rings, strongly regular rings, and S-weakly regular rings.
 

The objective of this paper is to initiate the study of rings in which each simple right R-module is flat, such rings will be called right SF-rings. Some important properties of right SF-rings are obtained. Among other results we prove that: If R is a semi prime ERT right SF-ring with zero socle, then R is a strongly regular ring.