Cilt:70 Sayı:02 (2021)

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The triple zero graph of a commutative ring
(Ankara Üniversitesi, 2021) Çelikel, Ece Yetkin; Other; Other
Let R be a commutative ring with non-zero identity. We define the set of triple zero elements of R by T Z ( R ) = { a ∈ Z ( R ) ∗ : there exists b , c ∈ R ∖ { 0 } such that a b c = 0 , a b ≠ 0 , a c ≠ 0 , b c ≠ 0 } . In this paper, we introduce and study some properties of the triple zero graph of R which is an undirected graph T Z Γ ( R ) with vertices T Z ( R ) , and two vertices a and b are adjacent if and only if a b ≠ 0 and there exists a non-zero element c of R such that a c ≠ 0 , b c ≠ 0 , and a b c = 0 . We investigate some properties of the triple zero graph of a general ZPI-ring R , we prove that d i a m ( T Z Γ ( R ) ) ∈ { 0 , 1 , 2 } and g r ( G ) ∈ { 3 , ∞ } .