Zero-Divisor Lattices

We define an equivalence relation on the set of zero-divisors to be that x is equivalent to y if the annihilators of x equal the annihilators of y. We also can put a partial order, ≤, on this relation defined to be that x ≤ y if and only if the annihilators of x is a subset of the annihilators of y.

We define < to be an order such that x < y if and only if x ≤ y and x ≠ y. In other words, for two vertices x and y in the zero-divisor lattice, we say that x < y if and only if the annihilators of x is a proper subset of the annihilators of y.

A Zero-Divisor Lattice, denoted Λ(R), is a lattice where the vertices are the equivalence classes of the nonzero zero-divisors as defined above, and there is an edge y → x if and only if x < y and there does not exist a z such that x < z < y.