Classification of nonlocal rings with genus one 3-zero-divisor hypergraphs

Selvakumar, K.; Ramanathan, Varun

doi:10.1080/00927872.2016.1206344

Cited by 16 publications

(2 citation statements)

References 19 publications

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“…As a notation, the set of all k-zero divisor elements of R is denoted by Z k (R). By associating an hypergraph to R, some properties of Z k (R) are determined in [2,5,6]. Also, the 3-zero divisor elements of R/I has been studied as 3-zero divisor elements of R with respect to an ideal I, [4].…”

Section: Introductionmentioning

confidence: 99%

The 3-Zero Divisor and Quasi 3-Zero Divisor Elements of a Commutative Ring with Respect to an Ideal

Sevim

Babaei

Ulucak

2023

Preprint

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For an ideal $I$ of a commutative ring $R$, we classify the set of all $3$-zero divisor elements of $R$ with respect to $I$. We prove that if $\bigcap_{i=1}^n P_i$ is a minimal prime decomposition of $\sqrt{I}$, then all elements of \begin{center} $\displaystyle{\bigcup_{i=1}^n P_i-\Big( \bigcup_{j=1}^n \Big\{x\in\bigcap_{i\in \{1,2,\ldots,n\}-\{j\}} P_i\backslash P_j: P_j\subseteq I:_Rx \Big\} \cup \sqrt{I}\Big)}$ \end{center} are $3$-zero divisor elements of $R$ with respect to $I$. Moreover, we determine $3$-zero divisors elements of $R$ with respect to $I$, when either $\sqrt{I}=I$ or $I$ is a primary ideal. Afterwards, we introduce quasi $3$-zero divisor element of $R$ with respect to $I$ and by using the minimal prime ideals of $R$ containing $I$ we characterize these elements. Moreover, we define quasi 3-zero divisor hypergraph of $R$ with respect to $I$ and we determine some properties of this hypergraph with the algebraic properties of $R$. 2000 Mathematics Subject Classification. 13A15, 13F30, 05C25.

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Section: Introductionmentioning

confidence: 99%

The 3-Zero Divisor and Quasi 3-Zero Divisor Elements of a Commutative Ring with Respect to an Ideal

Sevim

Babaei

Ulucak

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…The elements of are the vertices of , and those of the edges of . Consider that the edges ( , ) and ( , ) denote the same edge (For more information, see [3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%