The relationships between clean rings, r-clean rings, and f-clean rings

Abstract

Suppose R is a ring with identity. R called a regular ring if each element is a regular element. A ring R is called an r-clean ring if every element in R can be expressed as the sum of a regular element and an idempotent element. An element in the ring R is called clean if it can be expressed as the sum of a unit element and an idempotent element, so the ring R is called a clean ring if every element is clean. A ring of which each element is a full element is called an f-clean ring. We give some relationships between clean rings, regular rings, r-clean rings, and f-clean rings.