The Zero Divisor Graphs of Commutative Local Rings of Order p⁴ and p⁵

Abstract

To each commutative ring R we can associate a zero divisor graph whose vertices are the zero divisors of R and such that two vertices are adjacent if their product is zero. Detecting isomorphisms among zero divisor graphs can be reduced to the problem of computing the classes of R under a suitable semigroup congruence. Presently, we introduce a strategy for computing this quotient for local rings using knowledge about a generating set for the maximal ideal. As an example, we then compute Γ(R) for several classes of rings; with the results in [4] these classes include all local rings of order p ⁴ and p ⁵ for prime p.

Key Words:

• Finite commutative ring
• Finite local ring
• Zero divisor graph

2000 Mathematics Subject Classification:

• 05C25
• 05C30
• 13M05

Notes

Communicated by I. Swanson.