# Study of the rings in which each element express as the sum of an idempotent and pure

Document Type : Research Paper

Authors

1 College of Computer Science and Mathematics, Tikrit University, Tikrit, Salahaddin, Iraq

2 College of Education-Tuzkhurmatu, Tikrit University, Tuzkhurmatu, Salahaddin, Iraq

Abstract

This article aims to introduce the concept of $p-\mathrm{clean\ ring}\mathrm{s}$ as a generalization of some concepts such as $\mathrm{clean\ rings}$ and $r-\mathrm{clean\ rings}$. As the first result, we prove that every $\mathrm{clean\ ring}$ is a $p-\mathrm{clean\ ring}$ and every $r-\mathrm{clean\ ring}$ is a $p-\mathrm{clean\ ring}$. Furthermore, we give the relation between von Neumann $\mathrm{local\ ring}$ and $p-\mathrm{clean\ ring}$. Finally,  we investigate many properties of $p-\mathrm{clean\ ring}\mathrm{s}$.

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