ON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS

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REZAGHOLIBEIGI M.; NAGHIPOUR A. R.

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ISSN: 2588-4824

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Journal: JOURNAL OF ALGEBRAIC SYSTEMS Year:2019 | Volume:7 | Issue:1 Page(s): 51-68

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Title

ON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS

Author(s)

REZAGHOLIBEIGI M. | NAGHIPOUR A. R. | Issue Writer Certificate

Keywords

Rings

matrix rings

Jacobson radical

unit graphs

unitary Cayley graphs

spectrum

Abstract

Let R be a ring (not necessarily commutative) with nonzero identity. We define 􀀀 (R) to be the graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exist unit elements u; v of R such that x+uyv is a unit of R. In this paper, basic properties of 􀀀 (R) are studied. We investigate connectivity and the girth of 􀀀 (R), where R is a left Artinian ring. We also determine when the graph 􀀀 (R) is a cycle graph. We prove that if 􀀀 (R) = 􀀀 (Mn(F)) then R = Mn(F), where R is a ring and F is a finite field. We show that if R is a finite commutative semisimple ring and S is a commutative ring such that 􀀀 (R) = 􀀀 (S), then R = S. Finally, we obtain the spectrum of 􀀀 (R), where R is a finite commutative ring.

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APA: Copy

REZAGHOLIBEIGI, M., & NAGHIPOUR, A. R.. (2019). ON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS. JOURNAL OF ALGEBRAIC SYSTEMS, 7(1 ), 51-68. SID. https://sid.ir/paper/268349/en

Vancouver: Copy

REZAGHOLIBEIGI M., NAGHIPOUR A. R.. ON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS. JOURNAL OF ALGEBRAIC SYSTEMS[Internet]. 2019;7(1 ):51-68. Available from: https://sid.ir/paper/268349/en

IEEE: Copy

M. REZAGHOLIBEIGI, and A. R. NAGHIPOUR, “ON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS,” JOURNAL OF ALGEBRAIC SYSTEMS, vol. 7, no. 1 , pp. 51–68, 2019, [Online]. Available: https://sid.ir/paper/268349/en

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