Some results on the strongly annihilating-ideal graph of a commutative ring

Let R be a commutative ring with identity and A ( R ) be the set of ideals with non-zero annihilator. The strongly annihilating-ideal graph of R is defined as the graph SAG ( R ) with the vertex set A ( R ) ∗ = A ( R ) \ { 0 } and two distinct vertices I and J are adjacent if and only if I ∩ Ann ( J...

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Bibliographic Details
Published in:Boletín de la Sociedad Matemática Mexicana 2018-10, Vol.24 (2), p.307-318
Main Authors: Nikandish, R., Nikmehr, M. J., Tohidi, N. Kh
Format: Article
Language:English
Subjects:
Apexes
Mathematics
Mathematics and Statistics
Original Article
Rings (mathematics)
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Summary:Let R be a commutative ring with identity and A ( R ) be the set of ideals with non-zero annihilator. The strongly annihilating-ideal graph of R is defined as the graph SAG ( R ) with the vertex set A ( R ) ∗ = A ( R ) \ { 0 } and two distinct vertices I and J are adjacent if and only if I ∩ Ann ( J ) ≠ ( 0 ) and J ∩ Ann ( I ) ≠ ( 0 ) . We show that if R is a reduced ring with finitely many minimal primes, then SAG ( R ) is weakly prefect and an explicit formula for the vertex chromatic number of SAG ( R ) is given. Furthermore, strongly annihilating-ideal graphs with finite clique numbers are studied. Finally, we classify that all rings whose SAG ( R ) are planar.
ISSN:1405-213X
2296-4495
DOI:10.1007/s40590-017-0179-1