Maximal Ideals

In this chapter we consider the maximal ideals of incidence algebras. Before considering the maximal ideals of I(X, R), for a general commutative ring with identity R, we first focus our attention on incidence algebras over a field F. In the field case, the notion of a function being fully–nilpotent on a set simplifies. Indeed, if S ⊆ X and f ∈ I(X, F) is fully–nilpotent on S, then there is a positive integer n such that if x ₁ ≤ y ₁ ≤ x ₂ ≤ y ₂ ≤ … ≤ x_n ≤ y_n is any chain in S, then f(x_i,y_i ) = 0, for some 1 ≤ i ≤ n. In the field case, when f ∈ I(X,F) is fully–nilpotent on S, we will say that f is bounded on S .