All rings are assumed to be finite commutative with identity element. An element a \in R is called a regular element if there exists b \in R such that a=a2b, the element b is called a von Neumann inverse for a. A characterization is given for regular elements and their inverses in Zn, the ring of integers modulo n. The arithmetic function V(n), which counts the regular elements in Zn is studied. The relations between V(n) and Euler's phi-function j (n) are explored.

All rings are assumed to be finite commutative with identity element. An element a \in R is called a regular element if there exists b \in R such that a=a2b, the element b is called a von Neumann inverse for a. A characterization is given for regular elements and their inverses in Zn, the ring of integers modulo n. The arithmetic function V(n), which counts the regular elements in Zn is studied. The relations between V(n) and Euler's phi-function j (n) are explored.