%0 Journal Article
%T On Semi <em>¦Ð</em>-Regular Local Ring
%A Zubayda M. Ibraheem
%A Raghad A. Mustafa
%A Maha F. Khalf
%J Open Access Library Journal
%V 5
%N 10
%P 1-7
%@ 2333-9721
%D 2018
%I Open Access Library
%R 10.4236/oalib.1104788
%X <div style="text-align:justify;">
	<span style="line-height:1.5;">A ring </span><em style="line-height:1.5;">R</em><span style="line-height:1.5;"> is said to be a right (</span><span style="line-height:1.5;">left) semi </span><em style="line-height:1.5;">¦Ð</em><span style="line-height:1.5;">-regular local ring if and only if for all </span><em style="line-height:1.5;">a</em><span style="line-height:1.5;"> in </span><em style="line-height:1.5;">R</em><span style="line-height:1.5;">, either a or (1-a)  is a right (left) semi ¦Ð-regular element. The purpose of this paper is to give some characterization and properties of semi ¦Ð-regular local rings, and to study the relation between semi ¦Ð-regular local rings and local rings. From the main results of this work: 1) Let R be a semi ¦Ð-regular reduced ring. Then the idempotent associated element is unique. 2) Let R be a ring. Then R is a right semi ¦Ð-regular local ring if and only if either <em>r</em>(<em>a</em><em><sup>n</sup></em>) or <em>r</em>((1-<em>a</em>)<sup>n</sup>) is direct summand for all <em>a</em>¡Ê<em>R</em> and <em>n</em>¡Ê<em>Z</em><sup> </sup>. If R is a local ring with <em>r</em>(<em>a<sup>n</sup></em>)&nbsp;<em>r</em>(<em>a</em>) for all <em>a</em>¡Ê<em>R</em> and <em>n</em>¡Ê<em>Z</em><sup> </sup>, then <em>R</em> is a right semi <em>¦Ð</em>-regular local ring.</span> 
</div>
%K Local
%K Ring
%K Semi ¦Ð-Regular
%U http://www.oalib.com/paper/5298613