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

Document Type : Original Manuscript

Authors

Department of Mathematical Sciences, Shahrekord University, P.O.Box 115, Shahrekord, Iran.

Abstract

Let

R

be a ring (not necessarily commutative) with nonzero identity. We define

Γ (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

R

such that

x + u y v

is a unit of

R

. In this paper, basic properties of

Γ (R)

are studied. We investigate connectivity and the girth of

Γ (R)

, where

R

is a left Artinian ring. We also determine when the graph

Γ (R)

is a cycle graph. We prove that if

Γ (R) ≅ Γ (M_{n} (F))

then

R ≅ M_{n} (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

Γ (R) ≅ Γ (S)

, then

R ≅ S

. Finally, we obtain the spectrum of

Γ (R)

, where

R

is a finite commutative ring.

Keywords