Raf. J. of Comp. & Math’s. , Vol. 4, No. 1, 2007 57 On Rings whose Maximal Essential Ideals are Pure Raida D. Mahmood Awreng B. Mahmood raida.1961@uomosul.edu.iq awring2002@yahoo.com College of Computer sciences and Mathematics University of Mosul, Iraq Received on: 06/04/2006 Accepted on: 25/06/2006 ABSTRACT This paper introduces the notion of a right MEP-ring (a ring in which every maximal essential right ideal is left pure) with some of their basic properties; we also give necessary and sufficient conditions for MEP – rings to be strongly regular rings and weakly regular rings. Keywords: MEP-rings ,pure ideals ,weakly regular ring. نقي يكونساسيلي اعظمي ا فيها كل مثا التيحلقات ال حول ائ د. رحمودورنك بايز م احمودة داؤد م د اضيات ب والريلحاسو كلية علوم ا، موصلمعه ال جا ستلم:ريخ ال تا06 / 04 / 2006 ت اريخقبول ال: 25 / 06 / 2006 ملخص ال  دو  حلقدات مدحدوم ال ملحدوا الم ادل يقدMEP دح دهت الحلقداال( داله كدل جدثا م ا  ايمد أعظمد ه أساسد هقد ه اد و أيسد ر) ا وإعطد ا د رو الا عطد ا ل كد لساسد ية لحد ا الخدواس اي لحلقةلكافية ل ورية وا الضرMEP ظمة بضعف ة ومنظمة بقو حلقة من لكه تكون لمفتاحية:ت اكلما ال و حلقات م الMEP ، قه اله م، ظمة بضعف حلقة من 1- Introduction An ideal I of a ring R is said to be right (left)pure if for every I a  , there exists I b  such that a=ab (a=ba),[1],[2]. Throughout this paper, R is an associative ring with unity. Recall that: 1) R is called reduced if R has no non _zero nilpotent elements. 2) For any element a in R we define the right annihilator of a by r(a)={ 0 : =  ax R x } , and likewise the left annihilator l(a). 3) R is strongly regular [4], if for every R a  ,there exists R b  such that b a a 2 = .