Intersection graphs of graded ideals of graded rings

Tariq Alraqad; Hicham Saber; Rashid Abu-Dawwas; Tariq Alraqad; Hicham Saber; Rashid Abu-Dawwas

doi:10.3934/math.2021600

AIMS Mathematics

2021, Volume 6, Issue 10: 10355-10368. doi: 10.3934/math.2021600

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Research article

Intersection graphs of graded ideals of graded rings

1.
Department of Mathematics, Faculty of Science, University of Ha'il, Ha'il, Kingdom of Saudi Arabia
2.
Department of Mathematics, Yarmouk University, Irbid, Jordan

Received: 12 December 2020 Accepted: 12 July 2021 Published: 16 July 2021
MSC : 13A02, 16W502, 05C25

In this article, we introduce and study the intersection graph of graded ideals of a graded ring. The intersection graph of $ G- $graded ideals of a graded ring $ R $, denoted by $ Gr_G(R) $, is undirected simple graph defined on the set of nontrivial graded left ideals of $ R $, such that two left ideals are adjacent if their intersection is not trivial. We study properties for these graphs such as connectivity, regularity, completeness, domination numbers, and girth. We also present several results on the intersection graphs related to faithful grading, strong grading, and graded idealization.
- intersection graph,
- graded ring,
- graded ideal,
- homogeneous,
- strongly graded
Citation: Tariq Alraqad, Hicham Saber, Rashid Abu-Dawwas. Intersection graphs of graded ideals of graded rings[J]. AIMS Mathematics, 2021, 6(10): 10355-10368. doi: 10.3934/math.2021600

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Abstract

In this article, we introduce and study the intersection graph of graded ideals of a graded ring. The intersection graph of $ G- $graded ideals of a graded ring $ R $, denoted by $ Gr_G(R) $, is undirected simple graph defined on the set of nontrivial graded left ideals of $ R $, such that two left ideals are adjacent if their intersection is not trivial. We study properties for these graphs such as connectivity, regularity, completeness, domination numbers, and girth. We also present several results on the intersection graphs related to faithful grading, strong grading, and graded idealization.

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