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

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 (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.