Abstract/Details

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

O'Donnell, Christopher Jay.
University of Connecticut ProQuest Dissertations Publishing,  1992. 9315394.

### 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.

### Indexing (details)

Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences
Title
Maximal and minimal prime ideals of incidence algebras with applications to ring theory
Author
O'Donnell, Christopher Jay
Number of pages
79
Degree date
1992
School code
0056
Source
DAI-B 54/01, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
979-8-207-95397-7
University/institution
University of Connecticut
University location
United States -- Connecticut
Degree
Ph.D.
Source type
Dissertation or Thesis
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
9315394
ProQuest document ID
303990358