Necessary and Sufficient Conditions for the Embedding of Z_n into Z_m

This paper investigates the conditions required for the existence of an injective ring homomorphism (embedding) between cyclic rings (\mathbb{Z}_n, +_n, •_n) and (\mathbb{Z}_m, +_m, •_m). We prove that a necessary and sufficient condition for such an embedding is n\m and gcd(n, m/n) = 1. By leveraging the properties of idempotent elements and Bézout's Identity, we construct an explicit embedding mapping. Additionally, we propose an algorithmic approach to identify the local identity of the corresponding subring, simplifying the construction of the embedding through a direct identity shift e = (1+n) (mod m).