Upper dimension and bases of zero-divisor graphs of commutative rings

Abstract

For a commutative ring with non-zero zero divisor set , the zero divisor graph of is with vertex set , where two distinct vertices and are adjacent if and only if . The upper dimension and the resolving number of a zero divisor graph of some rings are determined. We provide certain classes of rings which have the same upper dimension and metric dimension and give an example of a ring for which these values do not coincide. Further, we obtain some bounds for the upper dimension in zero divisor graphs of commutative rings and provide a subset of vertices which cannot be excluded from any resolving set.

Keywords

• ring
• zero-divisor graph
• upper dimension
• resolving number