Necessary and Sufficient Conditions for a Finite Cyclic Group to be a Cyclic Module - Cahyadi, Arya Raditya; Sylviani, Sisilia; Gyenin, Mary (Indonesian Journal of Applied Mathematics and Statistics, 2025)
The structures of cyclic groups and cyclic modules are fundamental topics in abstract algebra. While their individual structures are well-established, the precise conditions that link them over an arbitrary ring  are often not detailed in the literature. This paper aims to find the necessary and sufficient conditions for a finite cyclic group  of order n to be a cyclic -module for an arbitrary ring . The method used in this research is by utilizing the fundamental theorem stating a relationship between an abelian group viewed as a -module and a ring homomorphism from  to the ring of endomorphism of its group, which is isomorphic to . The result of this research is a theorem proving that a finite cyclic group  of order  can be endowed with the structure of a cyclic -module if and only if there exists a surjective ring homomorphism from  to  (the ring of integers modulo ). This finding establishes a clear connection between these algebraic structures.