Author

# Anirudhdha Parmar

Bio: Anirudhdha Parmar is an academic researcher from Saurashtra University. The author has contributed to research in topics: Reduced ring & Maximal ideal. The author has an hindex of 1, co-authored 3 publications receiving 4 citations.

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01 Sep 2017TL;DR: Alilou, Amjadi and Sheikholeslami as discussed by the authors investigated a graph whose vertex set is the set of all nontrivial ideals of R and distinct vertices I, J are joined by an edge in this graph if and only if either ann(I)J = (0) or ann(J)I = (1).

Abstract: The rings considered in this article are commutative with identity which are not fields. Let R be a ring. A. Alilou, J. Amjadi and Sheikholeslami introduced and investigated a graph whose vertex set is the set of all nontrivial ideals of R and distinct vertices I, J are joined by an edge in this graph if and only if either ann(I)J = (0) or ann(J)I = (0). They called this graph as a new graph associated to a commutative ring.Their above mentioned work appeared in the Journal, Discrete Mathematics Algorithms and Applications. The aim of this article is to investigate the interplay between some graph- theoretic properties of the complement of a new graph associated to a commutative ring R and the ring -theoretic-properties of R.

2 citations

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TL;DR: The aim of this paper is to study the interplay between the graph-theoretic properties of [Formula: see text] and the ring-the theoretical properties of [/Formula]: see text.

Abstract: Let R be a commutative ring with identity which is not an integral domain. Let 𝔸(R) denote the set of all annihilating ideals of R and let us denote 𝔸(R)\{(0)} by 𝔸(R)∗. For an ideal I of R, we den...

1 citations

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TL;DR: In this article, the authors define the notion of maximal non-maximal ideal of a ring and a ring's proper ideal, i.e., a ring ideal that is maximal with respect to the property of not being a prime ideal.

Abstract: The rings considered in this article are commutative with identity $1neq 0$. By a proper ideal of a ring $R$, we mean an ideal $I$ of $R$ such that $Ineq R$. We say that a proper ideal $I$ of a ring $R$ is a maximal non-prime ideal if $I$ is not a prime ideal of $R$ but any proper ideal $A$ of $R$ with $ Isubseteq A$ and $Ineq A$ is a prime ideal. That is, among all the proper ideals of $R$, $I$ is maximal with respect to the property of being not a prime ideal. The concept of maximal non-maximal ideal and maximal non-primary ideal of a ring can be similarly defined. The aim of this article is to characterize ideals $I$ of a ring $R$ such that $I$ is a maximal non-prime (respectively, a maximal non maximal, a maximal non-primary) ideal of $R$.

1 citations

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TL;DR: In this article, the authors consider a simple graph associated with R denoted by ΩR∗, whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) =(0).

Abstract: Let R be a commutative ring with identity. In this paper, we consider a simple graph associated with R denoted by ΩR∗, whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) = (0). In this paper, we initiate the study of the graph ΩR∗ and we investigate its properties. In particular, we show that ΩR∗ is a connected graph with diam(ΩR∗) ≤ 3 unless R is isomorphic to a direct product of two fields. Moreover, we characterize all commutative rings R with at least two maximal ideals for which ΩR∗ are planar.

6 citations

01 Jan 2018

TL;DR: It is shown that ΩR∗ is a connected graph with diam(ΩR ∼3) ≤ 3 unless R is isomorphic to a direct product of two fields and all commutative rings R with at least two maximal ideals for which Ω R∗ are planar are characterized.

Abstract: In this paper some properties of the complement of a new graph associated with a commutative ring are investigated ....