### Keywords : strongly regular

##### On MP-Rings

Raida D. Mahmood; Azhar M. Hajo

AL-Rafidain Journal of Computer Sciences and Mathematics, 2020, Volume 14, Issue 1, Pages 35-40
DOI: 10.33899/csmj.2020.164797

An ideal I of a ring R is said to be right (left) Pure if for every  , there is    such that  . A ring R is said to be right (left) MP-ring, if every maximal right (left) ideal of R is a left (right) pure. In this paper have been studied some new properties of MP-rings, there connections with strongly regular rings.
Some of the main result of the present work are as follows:
1- Let R be aright MP-ring, r(a) is a W-ideal for all  then
a- Every essential ideal is a direct summand.
b- R is strongly regular ring.
2- Let R be aright MP-ring. If R is right almost abelian left NBF ring, then R is strongly regular.

##### On П – Pure Ideals

AL-Rafidain Journal of Computer Sciences and Mathematics, 2014, Volume 11, Issue 2, Pages 83-86
DOI: 10.33899/csmj.2014.163751

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.

##### On GP- Ideals

Raida D. Mahmood; Shahla M. Khalil

AL-Rafidain Journal of Computer Sciences and Mathematics, 2009, Volume 6, Issue 3, Pages 57-63
DOI: 10.33899/csmj.2009.163821

In this work we give some new properties of GP- ideals as well as the relation between GP- ideals, - π regular and simple ring. Also we consider rings with every principal ideal are GP- ideals and establish relation between such rings with strongly  regular and local rings.

##### Maximal Generalization of Pure Ideals

Raida D. Mahmood; Awreng B. Mahmood

AL-Rafidain Journal of Computer Sciences and Mathematics, 2008, Volume 5, Issue 1, Pages 21-27
DOI: 10.33899/csmj.2008.163946

The purpose of this paper is to study the class of the rings for which every maximal right ideal is left GP-ideal. Such rings are called MGP-rings and give some of their basic properties as well as the relation between
MGP-rings, strongly regular ring, weakly regular ring and kasch ring.

##### On Generalized Simple P-injective Rings

Nazar H. Shuker; Raida D. Mahmood

AL-Rafidain Journal of Computer Sciences and Mathematics, 2005, Volume 2, Issue 1, Pages 21-26
DOI: 10.33899/csmj.2005.164064

In this work, we study generalized simple P-injective rings. New properties of such rings are given, and a characterization of division rings and strongly regular rings in terms of GSP rings is obtained.