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Let R be a commutative ring and I its proper ideal, let S(I)be the set of all elements of R that are not prime to I. Here we introduce and study the total graph of a commutative ring R with respect to proper ideal I, denoted by T(ΓI (R)). It is the (undirected) graph with all ele-ments of R as vertices, and for distinct x; y 2 R, the vertices x and y are adjacent if and only if x + y 2 S(I). The total graph of a commutative ring, that denoted by T((R)), is the graph where the vertices are all elements of R and where there is an undirected edge between two distinct vertices x and y if and only if x + y 2 Z(R) which is due to Anderson and Badawi [2]. In the case I = f0g, T(ΓI (R)) = T(Γ(R)); this is an important result on the de nition.
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참고문헌 (8)
참고문헌 신청
A. Yousefian Darani / 2008 / Primal and weakly primal submodules and related results / 박사 / University of Guilan
B. Bollobas / 1979 / Graph Theory: An Introduction Course / Springer-Verlag
D. F. Anderson / 1999 / The zero-divisor graph of a commutative ring / J. Algebra 217 (2) : 434 ~ 447
D. F. Anderson / 2008 / The total graph of a commutative ring / J. Algebra 320 (7) : 2706 ~ 2719
L. Fuchs / 1950 / On primal ideals / Proc. Amer. Math. Soc 1 : 1 ~ 6
B. Bollobas / 1979 / Graph Theory: An Introduction Course / Springer-Verlag
D. F. Anderson / 1999 / The zero-divisor graph of a commutative ring / J. Algebra 217 (2) : 434 ~ 447
D. F. Anderson / 2008 / The total graph of a commutative ring / J. Algebra 320 (7) : 2706 ~ 2719
L. Fuchs / 1950 / On primal ideals / Proc. Amer. Math. Soc 1 : 1 ~ 6
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