Types of Irreducible Divisor Graphs of Noncommutative Domains, II
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917633093271552 |
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| author | Naser, A. Abdel-Khalek, R. E. Salem, R. M. Hassanein, A. M. |
| author_facet | Naser, A. Abdel-Khalek, R. E. Salem, R. M. Hassanein, A. M. |
| contents | In this paper, we continue investigation of the directed and undirected irreducible divisor graph concepts $G(x)$ and $Γ(x)$ of $x\in D^{\ast} \backslash U(D)$, respectively, which were introduced in [7]. Consequently, we introduce two generalizations of these concepts. The first one is the irreducible divisor simplicial complex $S(x)$ of $x\in D^{\ast} \backslash U(D)$ in a noncommutative atomic domain $D$, which simultaneously extends the commutative case that was introduced by R. Baeth and J. Hobson in [3]. The second one is the directed and undirected $τ$-irreducible divisor graphs $G_{τ}(x)$ and $Γ_{τ}(x)$ of $x\in D^{\ast} \backslash U(D)$, respectively, in a noncommutative $τ$-atomic domain $D$ with a symmetric and associate preserving relation $τ$ on $D^{\ast} \backslash U(D)$. Those graphs also extend the commutative case that was introduced by C. P. Mooney in [5]. Furthermore, we extend the results of [3] and [5] to give a characterization of n-unique factorization domains via those two generalizations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04873 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Types of Irreducible Divisor Graphs of Noncommutative Domains, II Naser, A. Abdel-Khalek, R. E. Salem, R. M. Hassanein, A. M. Rings and Algebras In this paper, we continue investigation of the directed and undirected irreducible divisor graph concepts $G(x)$ and $Γ(x)$ of $x\in D^{\ast} \backslash U(D)$, respectively, which were introduced in [7]. Consequently, we introduce two generalizations of these concepts. The first one is the irreducible divisor simplicial complex $S(x)$ of $x\in D^{\ast} \backslash U(D)$ in a noncommutative atomic domain $D$, which simultaneously extends the commutative case that was introduced by R. Baeth and J. Hobson in [3]. The second one is the directed and undirected $τ$-irreducible divisor graphs $G_{τ}(x)$ and $Γ_{τ}(x)$ of $x\in D^{\ast} \backslash U(D)$, respectively, in a noncommutative $τ$-atomic domain $D$ with a symmetric and associate preserving relation $τ$ on $D^{\ast} \backslash U(D)$. Those graphs also extend the commutative case that was introduced by C. P. Mooney in [5]. Furthermore, we extend the results of [3] and [5] to give a characterization of n-unique factorization domains via those two generalizations. |
| title | Types of Irreducible Divisor Graphs of Noncommutative Domains, II |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2404.04873 |