Rings which are generated by their units: a graph theoretical approach

Maimani, Hamid Reza; Pournaki, M. R.; Yassemi, Siamak

doi:10.4171/em/134

“…Later, more properties of the unit graph of a ring and its applications were given in [7,8,9]. Let us first define this notion.…”

Section: Preliminaries and The Statement Of The Main Resultsmentioning

confidence: 99%

Classification of rings with unit graphs having domination number less than four

Kiani¹,

Maimani²,

Pournaki³

et al. 2015

Rend. Sem. Mat. Univ. Padova

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Abstract. Let R be a finite commutative ring with nonzero identity. The unit graph of R is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x + y is a unit element of R. In this paper, a classification of finite commutative rings with nonzero identity in which their unit graphs have domination number less than four is given.

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“…The present paper deals with what is known as the unit graph of a ring, a notion that generalizes the idea of Grimaldi [1990] who introduced and studied in detail a graph G(‫ޚ‬ n ) in which the vertex set is the ring ‫ޚ‬ n of integers modulo a positive integer n, and two distinct vertices x and y are adjacent if and only if x + y is a unit in ‫ޚ‬ n . In general, given an arbitrary ring R with nonzero identity, its unit graph G(R) is defined to be the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x + y is a unit in R. Some of the properties of this graph have been studied in detail in [Ashrafi et al 2010;Maimani et al 2010a;2010b;2010c;2011b]. The graphs in Figure 1 are the unit graphs of the rings indicated.…”

Section: Introductionmentioning

confidence: 99%

Nonplanarity of unit graphs and classification of the toroidal ones

Das

¹

,

Maimani

²

,

Pournaki

³

et al. 2014

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The unit graph of a ring R with nonzero identity is the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x + y is a unit in R. In this paper, we derive several necessary conditions for the nonplanarity of the unit graphs of finite commutative rings with nonzero identity, and determine, up to isomorphism, all finite commutative rings with nonzero identity whose unit graphs are toroidal.

show abstract

“…The present paper deals with what is known as the unit graph of a ring, a notion that generalizes the idea of Grimaldi [1990] who introduced and studied in detail a graph G(‫ޚ‬ n ) in which the vertex set is the ring ‫ޚ‬ n of integers modulo a positive integer n, and two distinct vertices x and y are adjacent if and only if x + y is a unit in ‫ޚ‬ n . In general, given an arbitrary ring R with nonzero identity, its unit graph G(R) is defined to be the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x + y is a unit in R. Some of the properties of this graph have been studied in detail in [Ashrafi et al 2010;Maimani et al 2010a;2010b;2010c;2011b]. The graphs in Figure 1 are the unit graphs of the rings indicated.…”

Section: Introductionmentioning

confidence: 99%