On the problem of the finiteness of the compressed zero divisor graphs of Artinian rings

TL;DR

This paper investigates the conditions under which the clique number of compressed zero-divisor graphs of Artinian rings is finite, establishing finiteness for rings of length five, extending previous results.

Contribution

It proves that for Artinian rings of length five, the clique number of the associated compressed zero-divisor graph is finite, filling a gap between known cases.

Findings

Finiteness of clique number for length five rings.

Extension of previous results from lengths ≤4 and length 6.

Provides a complete characterization for length five rings.

Abstract

Let R be an Artinian ring and G be the compressed zero-divisor graph associated to R. The question of when the clique number of compressed zero-divisor graphs is finite was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S. Spiroff, in their survey paper entitled On Zero-divisor Graphs. They proved that if length of R is at most four then the clique number is finite. In the length six case they gave an example of a ring where the clique is infinite. In this paper we show that when length of ring is five then the clique number of compressed zero-divisor graph is finite.