5 research outputs found
Some results on the comaximal ideal graph of a commutative ring
The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let be a ring such that admits at least two maximal ideals. Recall from Ye and Wu (J. Algebra Appl. 11(6): 1250114, 2012) that the comaximal ideal graph of , denoted by is an undirected simple graph whose vertex set is the set of all proper ideals of such that , where is the Jacobson radical of and distinct vertices , are joined by an edge in if and only if . In Section 2 of this article, we classify rings such that is planar. In Section 3 of this article, we classify rings such that is a split graph. In Section 4 of this article, we classify rings such that is complemented and moreover, we determine the -vertices of
The exact annihilating-ideal graph of a commutative ring
The rings considered in this article are commutative with identity. For an ideal of a ring , we denote the annihilator of in by . An ideal of a ring is said to be an exact annihilating ideal if there exists a non-zero ideal of such that and . For a ring , we denote the set of all exact annihilating ideals of by and by . Let be a ring such that . With , in [Exact Annihilating-ideal graph of commutative rings, {\it J. Algebra and Related Topics} {\bf 5}(1) (2017) 27-33] P.T. Lalchandani introduced and investigated an undirected graph called the exact annihilating-ideal graph of , denoted by whose vertex set is and distinct vertices and are adjacent if and only if and . In this article, we continue the study of the exact annihilating-ideal graph of a ring. In Section 2 , we prove some basic properties of exact annihilating ideals of a commutative ring and we provide several examples. In Section 3, we determine the structure of , where either is a special principal ideal ring or is a reduced ring which admits only a finite number of minimal prime ideals
On Predicting lung cancer subtypes using ‘omic’ data from tumor and tumor-adjacent histologically-normal tissue
The exact annihilating-ideal graph of a commutative ring
The rings considered in this article are commutative with identity. For an ideal of a ring , we denote the annihilator of in by . An ideal of a ring is said to be an exact annihilating ideal if there exists a non-zero ideal of such that and . For a ring , we denote the set of all exact annihilating ideals of by and by . Let be a ring such that . With , in [Exact Annihilating-ideal graph of commutative rings, {\it J. Algebra and Related Topics} {\bf 5}(1) (2017) 27-33] P.T. Lalchandani introduced and investigated an undirected graph called the exact annihilating-ideal graph of , denoted by whose vertex set is and distinct vertices and are adjacent if and only if and . In this article, we continue the study of the exact annihilating-ideal graph of a ring. In Section 2 , we prove some basic properties of exact annihilating ideals of a commutative ring and we provide several examples. In Section 3, we determine the structure of , where either is a special principal ideal ring or is a reduced ring which admits only a finite number of minimal prime ideals