Ideal-based quasi cozero divisor graph of a commutative ring
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| Format: | Preprint |
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2024
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| _version_ | 1866914921563815936 |
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| author | Farshadifar, F. |
| author_facet | Farshadifar, F. |
| contents | Let R be a commutative ring with identity, and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by $Γ_I(R)$, is the graph whose vertices are the set $\{x \in R \setminus I | xy \in I$ for some $y \in R \setminus I\}$, where distinct vertices x and y are adjacent if and only if $xy \in I$. The cozero-divisor graph with respect to I, denoted by $Γ''_I(R)$, is the graph of $R$ with vertices $\{x \in R \setminus I | xR + I \neq R\}$, and two distinct vertices x and y are adjacent if and only if $x \notin yR + I$ and $y \notin xR + I$. In this paper, we introduce and investigate an undirected graph $QΓ''_I(R)$ of R with vertices $\{x \in R \setminus \sqrt{I} | xR + I \neq R$ and $xR + \sqrt{I} = xR + I\}$ and two distinct vertices x and y are adjacent if and only if $x \notin yR + I$ and $y \notin xR + I$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_13216 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ideal-based quasi cozero divisor graph of a commutative ring Farshadifar, F. Commutative Algebra Let R be a commutative ring with identity, and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by $Γ_I(R)$, is the graph whose vertices are the set $\{x \in R \setminus I | xy \in I$ for some $y \in R \setminus I\}$, where distinct vertices x and y are adjacent if and only if $xy \in I$. The cozero-divisor graph with respect to I, denoted by $Γ''_I(R)$, is the graph of $R$ with vertices $\{x \in R \setminus I | xR + I \neq R\}$, and two distinct vertices x and y are adjacent if and only if $x \notin yR + I$ and $y \notin xR + I$. In this paper, we introduce and investigate an undirected graph $QΓ''_I(R)$ of R with vertices $\{x \in R \setminus \sqrt{I} | xR + I \neq R$ and $xR + \sqrt{I} = xR + I\}$ and two distinct vertices x and y are adjacent if and only if $x \notin yR + I$ and $y \notin xR + I$. |
| title | Ideal-based quasi cozero divisor graph of a commutative ring |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2408.13216 |