The Inclusion Ideal Graph of Rings |
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| Authors: | S. Akbari M. Habibi A. Majidinya |
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| Affiliation: | 1. Department of Mathematical Sciences , Sharif University of Technology , Tehran , Iran;2. School of Mathematics , Institute for Research in Fundamental Sciences (IPM) , Tehran , Iran;3. Department of Mathematics , University of Tafresh , Tafresh , Iran;4. Department of Computer Sciences , Salman Farsi University of Kazerun , Kazerun , Iran |
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| Abstract: | Let R be a ring with unity. The inclusion ideal graph of a ring R, denoted by In(R), is a graph whose vertices are all nontrivial left ideals of R and two distinct left ideals I and J are adjacent if and only if I ? J or J ? I. In this paper, we show that In(R) is not connected if and only if R ? M 2(D) or D 1 × D 2, for some division rings, D, D 1 and D 2. Moreover, we prove that if In(R) is connected, then diam(In(R)) ≤3. It is shown that if In(R) is a tree, then In(R) is a caterpillar with diam(In(R)) ≤3. Also, we prove that the girth of In(R) belongs to the set {3, 6, ∞}. Finally, we determine the clique number and the chromatic number of the inclusion ideal graph for some classes of rings. |
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| Keywords: | Chromatic number Clique number Girth Inclusion ideal graph |
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