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On The Regular Elements In Z_N, Osama Alkam, Emad Abu Osba
On The Regular Elements In Z_N, Osama Alkam, Emad Abu Osba
Turkish Journal of Mathematics
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.