On the regular elements in Zn

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 = a^2b, the element b is called a von Neumann inverse for a. A characterization is given for regular elements and their inverses in Zn, the ringo f 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 (Varphi)(n) are explored.