Divisibility relation between the number of certain surjective group and ring homomorphisms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914051657826304 |
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| author | Kumar, Sonu Mandal, Priyabrata |
| author_facet | Kumar, Sonu Mandal, Priyabrata |
| contents | 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 $2 \cdot α$, where each prime factor $p$ of $α$ satisfies $p \equiv 3 \pmod{4}$. We further extend this result for finite abelian structures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_14266 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Divisibility relation between the number of certain surjective group and ring homomorphisms Kumar, Sonu Mandal, Priyabrata Commutative Algebra Combinatorics 2000: 20K01, 20K30, 11A41 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 $2 \cdot α$, where each prime factor $p$ of $α$ satisfies $p \equiv 3 \pmod{4}$. We further extend this result for finite abelian structures. |
| title | Divisibility relation between the number of certain surjective group and ring homomorphisms |
| topic | Commutative Algebra Combinatorics 2000: 20K01, 20K30, 11A41 |
| url | https://arxiv.org/abs/2502.14266 |