Group of units in a finite ring

Let R be a ring with nonzero identity. The unit graph of R, denoted by G R , has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this article, the basic properties of G R are investigated and some characterization results regarding connectedness, chromatic index, diameter, girth, and planarity of G R are given. (These terms are defined in Definitions and Remarks 4.1, 5.1, 5.3, 5.9, and 5.13.)

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“…Suppose that the unit graph G R is a cycle graph. By applying Corollary 4.5 of [5] we obtain that R is isomorphic to one of the following rings:…”

Section: Characterizations Of Certain Unit Graphsmentioning

confidence: 99%

“…By applying Corollary 4.5 of[5] we obtain that G R is a 2-regular graph if and only if R 2 × · · · × 2 × S, where S is one of the rings appeared in the statement of Theorem 3.2.The following result characterizes the unit graphs of rings that are complete graphs (cf. Definitions and Remarks 2.6).…”

mentioning

confidence: 95%

Unit Graphs Associated with Rings

Ashrafi

Maimani

Pournaki

et al. 2010

Communications in Algebra

129

100

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“…Theorem 2.3. [10,Cor. 4.4] Let G be a finite 2-group, and let R be a finite ring such that R × ∼ = G. Then, R is a direct product of rings, where every direct factor is isomorphic to one of the following rings:…”

Section: Preliminariesmentioning

confidence: 99%

Fuchs' problem for 2-groups

Swartz

Werner

2020

Journal of Algebra

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“…By applying Corollary 4.5 of [11] we obtain that G(R) is a 2-regular graph if and only if R ∼ = Z 2 × · · · × Z 2 × S, where S is one of the rings appearing in the statement of Theorem 3.4. The following theorem characterizes the complete bipartite unit graphs of rings.…”

Section: Theorem 34 ([6] Theorem 32) Let R Be a Finite Ring Then mentioning

confidence: 98%

Graphs Attached to Rings Revisited

et al. 2011

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In this paper, we discuss some recent results on graphs attached to rings. In particular, we deal with comaximal graphs, unit graphs, and total graphs. We then define the notion of cototal graph and, using this graph, we characterize the rings which are additively generated by their zero divisors. Finally, we glance at graphs attached to other algebraic structures. Mathematics Subject Classification (2010)

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Group of units in a finite ring

Cited by 14 publications

References 1 publication

Unit Graphs Associated with Rings

Unit Graphs Associated with Rings

Fuchs' problem for 2-groups

Graphs Attached to Rings Revisited

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