One-sided Commutativity of Nilpotent Elements at Zero

The concepts of left and right nilpotent reversible rings are defined and studied, which are closely related to the one-sided nilpotent structures of reversible rings, and are a proper subclass of CNZ rings. It is shown that a ring R is reversible if and only if R is semiprime and left nilpotent reversible ring. Various extension properties of left nilpotent reversible rings are studied, the following results are mainly proved:A ring R is left nilpotent reversible if and only if A(R,α) is left nilpotent reversible. If R is an Armendariz ring, then R is left nilpotent reversible if and only if Rx is left nilpotent reversible. If a right Ore ring R is a left nilpotent reversible ring, then its classical right quotient ring Q is left nilpotent reversible.