Necessary and Sufficient Conditions for a Finite Cyclic Group to be a Cyclic Module - Cahyadi, Arya Raditya; Sylviani, Sisilia; Gyenin, Mary (Lembaga Penelitian dan Pengembangan Matematika dan Statistika Terapan Indonesia, PT Anugrah Teknologi Kecerdasan Buatan PT Anugrah Teknologi Kecerdasan Buatan, 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.