The total graph of a commutative ring with respect to proper ideals
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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.
Abstract
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.
참고문헌 (8)
- S. Akbari, "The total graph and regular graph of a commutative ring", J. Pure Appl. Algebra, 213(12), pp.2224~2228 (2009)
- D. F. Anderson, "The total graph of a commutative ring", J. Algebra, 320(7), pp.2706~2719 (2008)
- D. F. Anderson, "The zero-divisor graph of a commutative ring", J. Algebra, 217(2), pp.434~447 (1999)
- M. F. Atiyah, "Introduction to Commutative Algebra" (1969), Addison- Wesley
- B. Bollobas, "Graph Theory: An Introduction Course" (1979), Springer-Verlag
- A. Yousefian Darani, "Primal and weakly primal submodules and related results", 박사, pp.2008
- L. Fuchs, "On primal ideals", Proc. Amer. Math. Soc, 1, pp.1~6 (1950)
- S. P. Redmond, "An ideal-based zero-divisor graph of a commutative ring", Comm. Algebra, 31(9), pp.4425~4443 (2003)