A Bipartite Graph Linking Units and Zero-Divisors

TL;DR

This paper introduces a new bipartite graph linking zero-divisors and units in a ring, exploring its properties and applications in characterizing finite reduced rings.

Contribution

It defines the bipartite zero-divisor--unit graph and studies its properties, providing a graphical characterization of finite reduced rings and distinguishing them from non-reduced rings.

Findings

The graph's properties vary with ring structure.

Complete invariance for finite reduced rings.

Graph distinguishes between reduced and non-reduced rings.

Abstract

Let $R$ be a commutative ring with identity. We introduce a novel bipartite graph $\mathcal{B}(R)$, the \textit{bipartite zero-divisor--unit graph}, whose vertex set is the disjoint union of the nonzero zero-divisors $Z(R)^*$ and the unit group $U(R)$. A vertex $z \in Z(R)^*$ is adjacent to $u \in U(R)$ if and only if $z + u \in Z(R)$. This construction provides an \textit{additive} counterpart to the well-established \textit{multiplicative} zero-divisor graphs. We investigate fundamental graph-theoretic properties of $\mathcal{B}(R)$, including connectedness, diameter, girth, chromatic number, and planarity. Explicit descriptions are given for rings such as $\mathbb{Z}_n$, finite products of fields, and local rings. Our results are sharpest for \textit{finite reduced rings}, where $\mathcal{B}(R)$ yields a graphical characterization of fields and serves as a complete invariant: $\mathcal{B}(R) \cong \mathcal{B}(S)$ implies $R \cong S$ for finite reduced rings $R$ and $S$. The graph also reveals structural distinctions between reduced and non-reduced rings, underscoring its utility in the interplay between ring-theoretic and combinatorial properties.