On Rings whose Maximal Essential Ideals are Pure

Raida Mahmood,Awreng Mahmood

doi:10.33899/csmj.2007.163995

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.

Highlights

5) A ring R is said to be semi-commutative if xy=0 implies that xRy=0,for all x,y R .It is easy to see that R is semi-commutative if and only if every right annihilator in R is a two-sided ideal [8]
2-MEP-Rings: we introduce the notion of a right MEP-ring with some of their basic properties; Definition 2.1: A ring R is said to be right MEP-ring if every maximal essential right ideal of R is left pure
Proof: By Theorem (2.2), R is a reduced ring .We show that RaR+r(a)=R, for any a R

Summary

Introduction

‫ حلقة من ظمة بضعف‬،‫ م اله قه‬،MEP ‫ الحلقات م و‬:‫الكلمات المفتاحية‬ 1- Introduction An ideal I of a ring R is said to be right (left)pure if for every a I , there exists b I such that a=ab (a=ba),[1],[2]. Proof: Let a be a non zero element of R, such that a2 = 0 and let M be a maximal right ideal containing r (a). Since R is a right MEP- ring, M is left pure for every a M.

Results

Conclusion