The Inclusion Ideal Graph of Rings

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.

Key Words:

• Chromatic number
• Clique number
• Girth
• Inclusion ideal graph

2010 Mathematics Subject Classification:

• 05C05
• 05C15
• 05C69
• 16D25

ACKNOWLEDGMENT

The authors would like to express their deep gratitude to the referee for a very careful reading of the paper, and many valuable comments, which have greatly improved the presentation of the paper.

Notes

Communicated by M. Bresar.