On ringswhose principal ideals are pure

Abstract EN

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 Introduction : 1.

Throughout this paper, R will denote associative ring with identity.

We recall that : 1) For any element a in R, we define the right annihilator of a by r (a) = {x ÎR : ax = 0}, and likewise the left annihilator l (a).

2) Y, Z, J will denote respectively the right singular ideal, left singular ideal and Jacobson radical of R.

3).

Following [2] a ring R is called reduced if R has no non–zero nilpotent elements 4) R is called uniform if every non–zero ideal of R is essential, see [3] 2- PIP–Rings Following [1], an ideal I is said to be a right (left) pure if for every a ÎI, there exists bÎI, such that a = ab (ba).

Definition 2.1 : A ring R is said to be a right PIP–ring, if every principal ideal is pure.