The radius of unit graphs of rings

Let $ R $ be a ring with nonzero identity. The unit graph of $ R $ is a simple graph whose vertex set is $ R $ itself and two distinct vertices are adjacent if and only if their sum is a unit of $ R $. In this paper, we study the radius of unit graphs of rings. We prove that there exists a ring $ R...

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Published inAIMS mathematics Vol. 6; no. 10; pp. 11508 - 11515
Main Authors Li, Zhiqun, Su, Huadong
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2021
Subjects
radius
ring extension
self-injective ring
unit graph
unit sum number
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ISSN2473-6988
2473-6988
DOI10.3934/math.2021667

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Abstract Let $ R $ be a ring with nonzero identity. The unit graph of $ R $ is a simple graph whose vertex set is $ R $ itself and two distinct vertices are adjacent if and only if their sum is a unit of $ R $. In this paper, we study the radius of unit graphs of rings. We prove that there exists a ring $ R $ such that the radius of unit graph can be any given positive integer. We also prove that the radius of unit graphs of self-injective rings are 1, 2, 3, $ \infty $. We classify all self-injective rings via the radius of its unit graph. The radius of unit graphs of some ring extensions are also considered.
AbstractList Let $ R $ be a ring with nonzero identity. The unit graph of $ R $ is a simple graph whose vertex set is $ R $ itself and two distinct vertices are adjacent if and only if their sum is a unit of $ R $. In this paper, we study the radius of unit graphs of rings. We prove that there exists a ring $ R $ such that the radius of unit graph can be any given positive integer. We also prove that the radius of unit graphs of self-injective rings are 1, 2, 3, $ \infty $. We classify all self-injective rings via the radius of its unit graph. The radius of unit graphs of some ring extensions are also considered.
Let R be a ring with nonzero identity. The unit graph of R is a simple graph whose vertex set is R itself and two distinct vertices are adjacent if and only if their sum is a unit of R. In this paper, we study the radius of unit graphs of rings. We prove that there exists a ring R such that the radius of unit graph can be any given positive integer. We also prove that the radius of unit graphs of self-injective rings are 1, 2, 3, ∞. We classify all self-injective rings via the radius of its unit graph. The radius of unit graphs of some ring extensions are also considered.
Author Li, Zhiqun
Su, Huadong
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Snippet Let $ R $ be a ring with nonzero identity. The unit graph of $ R $ is a simple graph whose vertex set is $ R $ itself and two distinct vertices are adjacent if...
Let R be a ring with nonzero identity. The unit graph of R is a simple graph whose vertex set is R itself and two distinct vertices are adjacent if and only if...
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SubjectTerms radius
ring extension
self-injective ring
unit graph
unit sum number
Title The radius of unit graphs of rings
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