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 Z_n, the ring of integers modulo n. The arithmetic function V(n), which counts the regular elements in Z_n is studied. The relations between V(n) and Euler's phi-function \varphi (n) are explored.
ALKAM, OSAMA and OSBA, EMAD ABU (2008) "On the regular elements in Z_n," Turkish Journal of Mathematics: Vol. 32: No. 1, Article 4. Available at: https://journals.tubitak.gov.tr/math/vol32/iss1/4