The k-maximal hypergraph of commutative rings

Abstract

Let R be a commutative ring with identity, $k\ge 2$ a fixed integer and $\mathcal {I}(R,k)$ be the set of all k-maximal elements in R. The k-maximal hypergraph associated with R, denoted by $\mathcal {H}^k(R)$, is a hypergraph with the vertex set $\mathcal {I}(R, k)$ and for distinct elements $a_1, a_2,\ldots , a_k$ in $\mathcal {I}(R, k)$ the set $\{a_1, a_2,\ldots , a_k\}$ is an edge of $\mathcal {H}^k(R)$ if and only if $\sum \nolimits _{i=1}^{k} Ra_{i}=R$ and for all $1\le j\le k$. In this paper, the connectedness, diameter and girth of $\mathcal {H}^k(R)$ are studied. Moreover, the regularity and coloring of $\mathcal {H}^k(R)$ are investigated. Among other things, we characterize all finite commutative rings R for which the k-maximal hypergraph $\mathcal {H}^k(R)$ is outerplanar and planar.