Commutative Rings R Whose C(𝔸𝔾(R)) Consist of Only Triangles

Abstract

Let R be a commutative ring with identity. The set R of all ideals of R is a bounded semiring with respect to ordinary addition, multiplication and inclusion of ideals. The zero-divisor graph of R is called the annihilating-ideal graph of R, denoted by R . We write for the set of graphs whose cores consist of only triangles. In this paper, the types of the graphs in that can be realized as either the zero-divisor graphs of bounded semirings or the annihilating-ideal graphs of commutative rings are determined. A necessary and sufficient condition for a ring R such that R ∈ is given. Finally, a complete characterization in terms of quotients of polynomial rings is established for finite rings R with R ∈ . Also, a connection between finite rings and their corresponding graphs is realized.

Maximal Ideal Graph of Commutative Rings

“…The zero divisor graph of commutative rings was first introduced by Beck in [1]. After that, many mathematicians studied such graphs [2][3][4][5].…”

More on the annihilator-ideal graph of a commutative ring

“…Therefore, one of the most popular and active area in algebraic combinatorics is study of graphs associated with rings. Papers in this field apply combinatorial methods to obtain algebraic results in ring theory (see for instance [1], [7], [13] and [15]). Moreover, for the most recent study in this direction see [6], [11] and [15].…”

“…Papers in this field apply combinatorial methods to obtain algebraic results in ring theory (see for instance [1], [7], [13] and [15]). Moreover, for the most recent study in this direction see [6], [11] and [15]. Throughout this paper, R denotes a unitary commutative ring which is not an integral domain.…”