Some Relations Between Clean, J-Clean and Nil Clean Rings

The ring 𝑅 is called clean if every element 𝑎 of 𝑅 can be written as 𝑎=𝑒+𝑢 where 𝑒 is an idempotent and 𝑢 is a unit. A ring is called 𝐽-clean provided that every element of it can be written as the sum of an idempotent and an element in it’s jacobson radical. 𝑅 is said to be nil clean ring if any 𝑎 ∈ 𝑅, has the form 𝑎=𝑒+𝑣 where 𝑒 is an idempotent and 𝑣 is a nilpotent element. Finally, 𝑅 is called 𝐽-nil clean in case every element of 𝑅 can be written as the sum of an idempotent and a nilpotent element of it’s jacobson radical. In this note we want to express some relashions between these classes of rings. we will discuss some classes of these ring and define a partial order on them.