The search for conditions under which the right or left ring of fractions has the properties of new algebraic concepts has always been of interest to researchers. Nil reversible rings, as a generalization of reversible rings, are among the new concepts that have been introduced and studied in recent years. For this reason, we look for conditions under which the right ring of fractions of a ring is nil reversible. In particular, we study the nil reversibility of the right ring of fractions of von Neumann rings and right Noetherian rings.
A ring R is called nil reversible if for any nilpotent element a of R and for any r of R, we have ar=0 if and only if ra=0. We affirm that if R is a von Neumann ring and X is a multiplicative subset of the central regular elements of R, then the nil reversibility of the rings R and RX^(-1) are equivalent. Furthermore, in this case, the ring R is reversible if and only if RX^(-1 ) is nil reversible. Finally, we affirm that the classical right ring of fractions S of the semiprime right Noetherian ring R is nil reversible if and only if R is nil reversible. Also, suppose R is a semiprime right Noetherian ring. In this case, we affirm that the reducibility of the classical right ring of fractions S, the reducibility of R and the nil reversibility of S are equivalent.