Ring Strong Clean And The Ring Distribution

In 1977, Nicholson introduced the concept of clean rings as follows: A ring R is said to be clean ring if every element of R is the sum of an idempotent element and an invertible element. Because every clean ring is an exchange ring, more and more algebra scholars are researching the property of clean rings. In 1999, Nicholson further introduced the concept of strong clean rings as follows: A ring R is said to be strongly clean if every element of R is the sum of an idempotent element and an invertible element that commute with each other.In the present paper, we mainly study the cleaness of semilocal rings, the strongly cleaness of some 3×3 matrix rings over a commutative local ring and the cleaness of distributive regular rings.In chapter two, we investigate the cleaness of semilocal rings by the relation between local rings and semilocal rings and prove that for an Abelian ring R, if the idempotents of R/radR can be lifted to R, then R is clean.In chapter three, we study the strongly cleaness of some 3×3 matrix rings over a commutative local ring and prove that if R is a commutative local ring, andthen TM₃(R) is a strongly clean ring.In chapter four, we prove that every distributive regular ring is strongly clean.