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AN IDEAL - BASED ZERO-DIVISOR GRAPH OF POSETS
Elavarasan, Balasubramanian,Porselvi, Kasi Korean Mathematical Society 2013 대한수학회논문집 Vol.28 No.1
The structure of a poset P with smallest element 0 is looked at from two view points. Firstly, with respect to the Zariski topology, it is shown that Spec(P), the set of all prime semi-ideals of P, is a compact space and Max(P), the set of all maximal semi-ideals of P, is a compact $T_1$ subspace. Various other topological properties are derived. Secondly, we study the semi-ideal-based zero-divisor graph structure of poset P, denoted by $G_I$ (P), and characterize its diameter.
On Comaximal Graphs of Near-rings
Dheena, Patchirajulu,Elavarasan, Balasubramanian Department of Mathematics 2009 Kyungpook mathematical journal Vol.49 No.2
Let N be a zero-symmetric near-ring with identity and let ${\Gamma}(N)$ be a graph with vertices as elements of N, where two different vertices a and b are adjacent if and only if <a> + <b> = N, where <x> is the ideal of N generated by x. Let ${\Gamma}_1(N)$ be the subgraph of ${\Gamma}(N)$ generated by the set {n ${\in}$ N : <n> = N} and ${\Gamma}_2(N)$ be the subgraph of ${\Gamma}(N)$ generated by the set $N{\backslash}{\upsilon}({\Gamma}_1(N))$, where ${\upsilon}(G)$ is the set of all vertices of a graph G. In this paper, we completely characterize the diameter of the subgraph ${\Gamma}_2(N)$ of ${\Gamma}(N)$. In addition, it is shown that for any near-ring, ${\Gamma}_2(N){\backslash}M(N)$ is a complete bipartite graph if and only if the number of maximal ideals of N is 2, where M(N) is the intersection of all maximal ideals of N and ${\Gamma}_2(N){\backslash}M(N)$ is the graph obtained by removing the elements of the set M(N) from the vertices set of the graph ${\Gamma}_2(N)$.
AN IDEAL-BASED ZERO-DIVISOR GRAPH OF 2-PRIMAL NEAR-RINGS
Dheena, Patchirajulu,Elavarasan, Balasubramanian Korean Mathematical Society 2009 대한수학회보 Vol.46 No.6
In this paper, we give topological properties of collection of prime ideals in 2-primal near-rings. We show that Spec(N), the spectrum of prime ideals, is a compact space, and Max(N), the maximal ideals of N, forms a compact $T_1$-subspace. We also study the zero-divisor graph $\Gamma_I$(R) with respect to the completely semiprime ideal I of N. We show that ${\Gamma}_{\mathbb{P}}$ (R), where $\mathbb{P}$ is a prime radical of N, is a connected graph with diameter less than or equal to 3. We characterize all cycles in the graph ${\Gamma}_{\mathbb{P}}$ (R).
A GENERALIZED IDEAL BASED-ZERO DIVISOR GRAPHS OF NEAR-RINGS
Dheena, Patchirajulu,Elavarasan, Balasubramanian Korean Mathematical Society 2009 대한수학회논문집 Vol.24 No.2
In this paper, we introduce the generalized ideal-based zero-divisor graph structure of near-ring N, denoted by $\widehat{{\Gamma}_I(N)}$. It is shown that if I is a completely reflexive ideal of N, then every two vertices in $\widehat{{\Gamma}_I(N)}$ are connected by a path of length at most 3, and if $\widehat{{\Gamma}_I(N)}$ contains a cycle, then the core K of $\widehat{{\Gamma}_I(N)}$ is a union of triangles and rectangles. We have shown that if $\widehat{{\Gamma}_I(N)}$ is a bipartite graph for a completely semiprime ideal I of N, then N has two prime ideals whose intersection is I.
An ideal-based zero-divisor graph of 2-primal near-rings
Patchirajulu Dheena,Balasubramanian Elavarasan 대한수학회 2009 대한수학회보 Vol.46 No.6
In this paper, we give topological properties of collection of prime ideals in 2-primal near-rings. We show that Spec(N), the spectrum of prime ideals, is a compact space, and Max(N), the maximal ideals of N, forms a compact T_1-subspace. We also study the zero-divisor graph Γ_(I) (R) with respect to the completely semiprime ideal I of N. We show that Γ_(P)(R), where P is a prime radical of N, is a connected graph with diameter less than or equal to 3. We characterize all cycles in the graph Γ_(P)(R). In this paper, we give topological properties of collection of prime ideals in 2-primal near-rings. We show that Spec(N), the spectrum of prime ideals, is a compact space, and Max(N), the maximal ideals of N, forms a compact T_1-subspace. We also study the zero-divisor graph Γ_(I) (R) with respect to the completely semiprime ideal I of N. We show that Γ_(P)(R), where P is a prime radical of N, is a connected graph with diameter less than or equal to 3. We characterize all cycles in the graph Γ_(P)(R).
Poset Properties Determined by the Ideal - Based Zero-divisor Graph
Porselvi, Kasi,Elavarasan, Balasubramanian Department of Mathematics 2014 Kyungpook mathematical journal Vol.54 No.2
In this paper, we study some properties of finite or infinite poset P determined by properties of the ideal based zero-divisor graph properties $G_J(P)$, for an ideal J of P.
z<sup>J</sup>-Ideals and Strongly Prime Ideals in Posets
John, Catherine Grace,Elavarasan, Balasubramanian Department of Mathematics 2017 Kyungpook mathematical journal Vol.57 No.3
In this paper, we study the notion of $z^J$ - ideals of posets and explore the various properties of $z^J$-ideals in posets. The relations between topological space on Sspec(P), the set $I_Q=\{x{\in}P:L(x,y){\subseteq}I\text{ for some }y{\in}P{\backslash}Q\}$ for an ideal I and a strongly prime ideal Q of P and $z^J$-ideals are discussed in poset P.
Strongly Prime Ideals and Primal Ideals in Posets
John, Catherine Grace,Elavarasan, Balasubramanian Department of Mathematics 2016 Kyungpook mathematical journal Vol.56 No.3
In this paper, we study and establish some interesting results of ideals in a poset. It is shown that for a nonzero ideal I of a poset P, there are at most two strongly prime ideals of P that are minimal over I. Also, we study the notion of primal ideals in a poset and the relationship among the primal ideals and strongly prime ideals is considered.