On $ϕ$-Prüfer like conditions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929529397706752 |
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| author | Anebri, Adam Mahdou, Najib Oubouhou, El Houssaine |
| author_facet | Anebri, Adam Mahdou, Najib Oubouhou, El Houssaine |
| contents | In this paper, we investigate the question of when a $ϕ$-ring is $ϕ$-Prüfer using two types of techniques: first, by analysing the lattice structure of the nonnil ideals of $ϕ$-rings; and secondly, by considering content ideal techniques which were developed to study Gaussian polynomials. In particular, we conclude that every Gaussian $ϕ$-ring is $ϕ$-Prüfer. Key concepts such as $ϕ$-weak global dimension, primary ideals and irreducible ideals are discussed, along with their hereditary properties in $ϕ$-Prüfer rings. We also prove that any semi-local $ϕ$-Prüfer ring is a $ϕ$-Bézout ring. This paper includes several theorems and examples that provide insights into the $ϕ$-Prüfer rings and their implications in the field of ring theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04181 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $ϕ$-Prüfer like conditions Anebri, Adam Mahdou, Najib Oubouhou, El Houssaine Commutative Algebra In this paper, we investigate the question of when a $ϕ$-ring is $ϕ$-Prüfer using two types of techniques: first, by analysing the lattice structure of the nonnil ideals of $ϕ$-rings; and secondly, by considering content ideal techniques which were developed to study Gaussian polynomials. In particular, we conclude that every Gaussian $ϕ$-ring is $ϕ$-Prüfer. Key concepts such as $ϕ$-weak global dimension, primary ideals and irreducible ideals are discussed, along with their hereditary properties in $ϕ$-Prüfer rings. We also prove that any semi-local $ϕ$-Prüfer ring is a $ϕ$-Bézout ring. This paper includes several theorems and examples that provide insights into the $ϕ$-Prüfer rings and their implications in the field of ring theory. |
| title | On $ϕ$-Prüfer like conditions |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2410.04181 |