Implements of Graph Blow-Up in Co-Maximal Ideal Graphs

Abstract

In this article, graph blow-up is used to study the implements of the co-maximal graph 2 R and the co-maximal ideal graph R of a commutative ring R. We show that for a ring R with finitely many maximal ideals, the graph R is a blow-up of a Boolean graph, and we give more detailed structure of the graphs 2 R and R as blow-ups of Boolean graphs.We determine for what rings R, R or 2 R is a finite graph, respectively. In the end, we get some new results for rings by taking advantage of the graph 2 R .

Comaximal graph of $C(X)$

“…In [27], comaximal graph of a commutative ring was defined. On later, in [21], [26], [16], [28], [24], [22], [19], [2], [3], [1], [30], and [23], this investigation was continued.…”

Complemented graphs and blow-ups of Boolean graphs, with applications to co-maximal ideal graphs

“…The induced subgraph of G T on T x is a discrete graph, i.e., a graph without any edge, while for distinct vertices x, y of G, each vertex of T x is adjacent to all vertices of T y in G T if and only if x is adjacent to y in G, see [8,16,17] for details. The previous work shows that graph blow-up plays an essential role in the theory of the co-maximal ideal graph of a ring, see [21,22] for the concise definitions, the history, the recent development, and a list of references. This paper is organized in the following way.…”