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

 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}.

Keywords:

ring , clean ring , r-clean ring , local ring and regular ring