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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.19197 |
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| _version_ | 1866917163297669120 |
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| author | Maassarani, Mohamad |
| author_facet | Maassarani, Mohamad |
| contents | Given two seprable irreducible polynomials $P_1$ and $P_2$ over a filed $\mathbb{K}$. We show that the rings $\mathbb{K}[X]/(P_1^n)$ and $\mathbb{K}[X]/(P_2^n)$ are isomorphic if and only if their residue fields $\mathbb{K}[X]/(P_1)$ and $\mathbb{K}[X]/(P_2)$ are isomorphic. Partial results in this direction are obtained for the case where the polynomials are not seprable. We note that, given a seprable irreducible polynomial $P$, we prove that we have an isomorphism between $\mathbb{K}[X]/(P^n)$ and $(\mathbb{K}[X](P))[Y]/(Y^n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19197 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On some local rings Maassarani, Mohamad Commutative Algebra Given two seprable irreducible polynomials $P_1$ and $P_2$ over a filed $\mathbb{K}$. We show that the rings $\mathbb{K}[X]/(P_1^n)$ and $\mathbb{K}[X]/(P_2^n)$ are isomorphic if and only if their residue fields $\mathbb{K}[X]/(P_1)$ and $\mathbb{K}[X]/(P_2)$ are isomorphic. Partial results in this direction are obtained for the case where the polynomials are not seprable. We note that, given a seprable irreducible polynomial $P$, we prove that we have an isomorphism between $\mathbb{K}[X]/(P^n)$ and $(\mathbb{K}[X](P))[Y]/(Y^n)$. |
| title | On some local rings |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2512.19197 |