Intersection graphs of ideals of rings

On cycles in intersection graphs of rings

‎Let $R$ be a commutative ring with non-zero identity. ‎We describe all $C_3$‎- ‎and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring. ‎Also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎Finally, ‎we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs. ...

on cycles in intersection graphs of rings

‎let $r$ be a commutative ring with non-zero identity. ‎we describe all $c_3$‎- ‎and $c_4$-free intersection graph of non-trivial ideals of $r$ as well as $c_n$-free intersection graph when $r$ is a reduced ring. ‎also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎finally, ‎we shall prove that almost all artin rings $r$ have hamiltonian intersection graphs. ...

Some Properties of the Nil-Graphs of Ideals of Commutative Rings

Let R be a commutative ring with identity and Nil(R) be the set of nilpotent elements of R. The nil-graph of ideals of R is defined as the graph AG_N(R) whose vertex set is {I:(0)and there exists a non-trivial ideal  such that  and two distinct vertices  and  are adjacent if and only if . Here, we study conditions under which  is complete or bipartite. Also, the independence number of  is deter...

Ring graphs and complete intersection toric ideals

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideal...

Separative Ideals of Exchange Rings