## Maximal and minimal prime ideals of incidence algebras with applications to ring theory

### Abstract (summary)

Let X be a locally finite partially ordered set and R a commutative ring with identity. The incidence algebra of X over R is $I(X,R)=\{f : X\times X \to R \mid f(x,y) = 0$ if $x\not\leq y\}.$

We define a map C from the incidence algebra to X $\times$ Spec(R) by $C(f)=\{(x, P) \mid f(x,x)\in P\}$. Letting ${\cal L}$ be the set of finite intersection of images of elements of the incidence algebra, ${\cal L}$ is a distributive lattice under the operations of union and intersection. We show that C is a homeomorphism from Maxspec($I(X,R$)) to the Stone space of ${\cal L}$.

We also determine necessary and sufficient conditions for C to be a homeomorphism from Minspec($I(X,R$)) to the space of minimal prime filters on ${\cal L}$ and show the equivalence of: (i) $I(X,R)$ has finite Krull dimension, (ii) $I(X,R)$ has Krull dimension zero, and (iii) C is a homeomorphism from Spec($I(X,R))$ to the space of prime filters on ${\cal L}$. Finally, these results are used to obtain results on the product of commutative rings with identity.