The dimension graph of a commutative ring

Abstract

This paper introduces a simple graph associated to a commutative ring. Nonzero ideals I and J of a commutative ring R are called Krull dimension-dependent whenever dim R / ( I + J ) = min { dim R / I , dim R / J } . Based on this, we introduce the dimension graph of R. We study and investigate this graph, and we determine all rings with disconnected dimension graph. Moreover, we introduce strongly Krull dimension-dependent ideals, and we examine these ideals using a subgraph of the dimension graph.