Author

# P. Subbulakshmi

Bio: P. Subbulakshmi is an academic researcher from Manonmaniam Sundaranar University. The author has contributed to research in topics: Commutative ring & Principal ideal ring. The author has an hindex of 2, co-authored 3 publications receiving 7 citations.

##### Papers

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TL;DR: The commutative Artinian non-local ring R for which ΩR∗ has genus one and crosscap one is characterized, whose vertex set is the set of all non-trivial ideals of R.

Abstract: Let R be a commutative ring with identity. We consider a simple graph associated with R, denoted by ΩR∗, whose vertex set is the set of all non-trivial ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) = (0). In this paper, we characterize the commutative Artinian non-local ring R for which ΩR∗ has genus one and crosscap one.

5 citations

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TL;DR: In this paper, the planarity and genus of a simple undirected Artin ring R for which the genus of A G N (R) is either zero or one is characterized.

2 citations

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TL;DR: Some properties of nil-graph of ideals concerning connectedness, split and claw free, and all commutative Artinian rings for which the nil- graph has genus 2 are discussed.

Abstract: Let R be a commutative ring with identity and Nil(R) be the ideal consisting of all nilpotent elements of R. Let I(R)={I:I is a non-trivial ideal of R and there exists a non-trivial ideal J such t...

1 citations

##### Cited by

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TL;DR: The commutative Artinian non-local ring R for which ΩR∗ has genus one and crosscap one is characterized, whose vertex set is the set of all non-trivial ideals of R.

Abstract: Let R be a commutative ring with identity. We consider a simple graph associated with R, denoted by ΩR∗, whose vertex set is the set of all non-trivial ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) = (0). In this paper, we characterize the commutative Artinian non-local ring R for which ΩR∗ has genus one and crosscap one.

5 citations

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01 Dec 2018TL;DR: In this paper, all commutative Artinian nonlocal rings with genus one have genus one and the annihilating-ideal graph is defined as the graph with the vertices of two distinct vertices that are adjacent if and only if their genus is genus one.

Abstract: Let $R$ be a non-domain commutative ring with identity and $A^*(R)$ be theset of non-zero ideals with non-zero annihilators. We call an ideal $I$ of $R$, anannihilating-ideal if there exists a non-zero ideal $J$ of $R$ such that $IJ = (0)$.The annihilating-ideal graph of $R$ is defined as the graph $AG(R)$ with the vertexset $A^*(R)$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ =(0)$. In this paper, we characterize all commutative Artinian nonlocal rings $R$for which $AG(R)$ has genus one.

3 citations

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TL;DR: In this paper, the authors characterize all Artinian commutative nonlocal rings R with genus one k-annihilating-ideal hypergraphs and show that all of them have genus one vertices.

Abstract: Let R be a commutative ring and k an integer greater than 2 and let $${\cal A}(R, k)$$
be the set of all k-annihilating-ideals of R. The k-annihilating-ideal hypergraph of R, denoted by $${\cal A}{{\cal G}_k}(R)$$
, is a hypergraph with vertex set $${\cal A}(R, k)$$
, and for distinct elements I1, …, Ik in $${\cal A}(R, k)$$
, the set {I1, I2, …, Ik} is an edge of $${\cal A}{{\cal G}_k}(R)$$
if and only if $$\prod\limits_{i = 1}^k {{I_i} = (0)} $$
and the product of any (k − 1) elements of the set {I1, I2,…, Ik} is nonzero. In this paper, we characterize all Artinian commutative nonlocal rings R whose $${\cal A}{{\cal G}_3}(R)$$
has genus one.

3 citations

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01 Dec 2021TL;DR: The annihilator-inclusion ideal graph of R, denoted by ξR, is a graph whose vertex set is the of all non-zero proper ideals of R, and two distinct vertices $I$ and $J$ are adjacent if and only if either ANN(I) ⊆ J or ANN(J)⊆ I.

Abstract: Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R , denoted by ξR, is a graph whose vertex set is the of allnon-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacentif and only if either Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basicproperties of the graph ξR. In particular, we showthat ξR is a connected graph with diameter at most three, andhas girth 3 or ∞. Furthermore, we determine all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.

2 citations