<![CDATA[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.]]>
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