Divisibility relation between the number of certain surjective group and ring homomorphisms

Abstract

In this article, we identify the existence of a divisibility relationship between the number of ring homomorphisms and surjective group homomorphisms. We demonstrate that for finite cyclic structures, the number of ring homomorphisms from $\mathbb{Z}_m$ to $\mathbb{Z}_n$ is a divisor of the number of surjective group homomorphisms from $\mathbb{Z}_m$ to $\mathbb{Z}_n$, where $n$ is not of the form $2 \cdot \alpha$, where each prime factor $p$ of $\alpha$ satisfies $p \equiv 3 \pmod{4}$. We further extend this result for finite abelian structures.