On the comaximal ideal graph of a commutative ring

Abstract

Let R be a commutative ring with identity. We use Γ(R) to denote the comaximal ideal graph. The vertices of Γ(R) are proper ideals of R that are not contained in the Jacobson radical of R , and two vertices I and J are adjacent if and only if I + J = R. In this paper we show some properties of this graph together with the planarity and perfection of Γ(R) .

A note on comaximal ideal graph of commutative rings

“…In 2016, Azadi et al [8] studied the comaximal ideal graph and determined conditions for the distance between two vertices is 1 or 2 or 3. By using these results one can conclude the central sets.…”

“…In 2016, Azadi et al [8] studied the graph structure defined by M. Ye and T. Wu. They investigated the planarity and perfection of comaximal ideal graph.…”

Finite commutative rings whose line graphs of comaximal graphs have genus at most two

“…The cross research on rings and graphs has attracted lots of attention by many mathematicians. The definition of graph on a ring one based on the special elements of the ring, for example, zero-divisor graphs [6], unit graphs [3] and total graphs [1] of rings; another one based on the ideals of the ring, for example, comaximal ideal graph [4,5,19] and zerodivisor graphs with respect to ideals [10] of rings. The comaximal graphs of rings based on both elements and ideals of rings seems to be more interesting.…”

Coloring of zero-divisor graphs of posets and applications to graphs associated with algebraic structures

“…For instance, it is known that the class of chordal graphs is perfect; see Dirac [17]. The notion of perfectness, weakly perfectness and chordalness of graphs associated with algebraic structures has been an active area of research; see [1], [5], [7], [8], [15], [38], [39], [42], etc.…”