The k-maximal hypergraph of commutative rings

Let R be a commutative ring with identity, k ≥ 2 a fixed integer and I ( R , k ) be the set of all k -maximal elements in R . The k -maximal hypergraph associated with R , denoted by H k ( R ) , is a hypergraph with the vertex set I ( R , k ) and for distinct elements a 1 , a 2 , … , a k in I ( R ,...

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Bibliographic Details
Published in:Beiträge zur Algebra und Geometrie 2020-12, Vol.61 (4), p.747-757
Main Authors: Selvakumar, K., Amritha, V. C.
Format: Article
Language:English
Subjects:
Algebra
Algebraic Geometry
Convex and Discrete Geometry
Geometry
Graph theory
Mathematics
Mathematics and Statistics
Original Paper
Rings (mathematics)
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Summary:Let R be a commutative ring with identity, k ≥ 2 a fixed integer and I ( R , k ) be the set of all k -maximal elements in R . The k -maximal hypergraph associated with R , denoted by H k ( R ) , is a hypergraph with the vertex set I ( R , k ) and for distinct elements a 1 , a 2 , … , a k in I ( R , k ) the set { a 1 , a 2 , … , a k } is an edge of H k ( R ) if and only if ∑ i = 1 k R a i = R and for all 1 ≤ j ≤ k . In this paper, the connectedness, diameter and girth of H k ( R ) are studied. Moreover, the regularity and coloring of H k ( R ) are investigated. Among other things, we characterize all finite commutative rings R for which the k -maximal hypergraph H k ( R ) is outerplanar and planar.
ISSN:0138-4821
2191-0383
DOI:10.1007/s13366-020-00505-8