AL-Rafidain Journal of Computer Sciences and Mathematics

As a generalization of right pure ideals, we introduce the notion of right П – pure ideals. A right ideal I of R is said to be П – pure, if for every a Î I there exists b Î I and a positive integer n such that aⁿ ≠ 0 and  aⁿ b = aⁿ. In this paper, we give some characterizations and properties of П – pure ideals and it is proved that:
If every principal right ideal of a ring R is П – pure then,   
a).L (aⁿ) = L (aⁿ⁺¹) for every a Î R and for some  positive integer n .
b). R is directly finite ring.
 c). R is strongly   П – regular ring.
    
 

In this work, we study rings whose every principal ideal is a right pure. We give some properties of right PIP – rings and the connection between such rings and division rings.