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 $2\cdot \hat{I}\pm$, where each prime factor $p$ of $\hat{I}\pm$ satisfies $p\equiv 3(\mathrm{mod}4)$. We further extend this result for finite abelian structures.

10 pages