University of Guilan Journal of Algebra and Related Topics 2345-3931 9 1 2021 06 01 P-Regular and P-Local Rings 1 19 4804 10.22124/jart.2021.17609.1230 EN H. Ibrahim Hakmi Department of Mathematics Faculty of Sciences Damascus University, Syria Journal Article 2020 09 09 This paper is a continuation of study rings relative to right<br />ideal, where we study the concepts of regular and local rings<br />relative to right ideal. We give some relations between $P-$local<br />($P-$regular) and local (regular) rings. New characterization<br />obtained include necessary and sufficient conditions of a ring $R$<br />to be regular, local ring in terms $P-$regular, $P-$local of<br />matrices ring $M_{2}(R)$. Also, We proved that every ring is local<br />relative to any maximal right ideal of it. P-idempotents Regular and P-regular rings Local and P-local rings semi-potent rings https://jart.guilan.ac.ir/article_4804_6c28fdeaefbc161e7d6a1c88a4aaa721.pdf