On the regular elements in $Bbb{Z}_n$

Abstract

All rings are assumed to be finite commutative with identity element. An element a $epsilon$ R is called a regular element if there exists b $epsilon$ 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 $Bbb{Z}_n$, the ring of integers modulo n. The arithmetic function V (n), which counts the regular elements in $Bbb{Z}_n$ is studied. The relations between V(n) and Euler's phi-function $varphi(n)$ are explored.