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# Prime ideals in two-dimensional Noetherian domains and fiber products and connected sums

#### Abstract

This thesis concerns three topics in commutative algebra: 1) The projective line over the integers (Chapter 2), 2) Prime ideals in two-dimensional quotients of mixed power series-polynomial rings (Chapter 3), 3) Fiber products and connected sums of local rings (Chapter 4). ^ In the first chapter we introduce basic terminology used in this thesis for all three topics. ^ In the second chapter we consider the partially ordered set (poset) of prime ideals of the projective line Proj( Z [h, k]) over the integers Z , and we interpret this poset as Spec( Z [*x*]) ∪ Spec( Z1x ) with an appropriate identification. ^ We have some new results that support Aihua Li and Sylvia Wiegand's conjecture regarding the characterization of Proj( Z [h, k]). In particular we show that a possible axiom for Proj( Z [h, k]) proposed by Arnavut, Li and Wiegand holds for some previously unknown cases. ^ We study the sets of prime ideals of polynomial rings, power series rings and mixed power series-polynomial rings in Chapter 3. Let *R* be a one-dimensional Noetherian domain and let *x* and * y* be indeterminates. We describe the prime spectra of certain two-dimensional quotients of mixed power series/polynomial rings over *R*, that is, Spec( Rx yQ ) and Spec( Ry xQ^{′ } ), where *Q* and *Q'* are certain height-one prime ideals of *R*[[*x*]][*y*] and *R*[*y*][[*x*]] respectively. ^ In the last chapter we describe some ring-theoretic and homological properties of fiber products and connected sums of local rings. For Gorenstein Artin * k*-algebras *R* and *S* where * k* is a field, the connected sum, *R# _{k}S*, is a quotient of the classical fiber product. We give basic properties of connected sums over a field and show that certain Gorenstein local

*k*-algebras decompose as connected sums. We generalize structure theorems given by Sally, Elias and Rossi that show two types of Gorenstein local

*k*-algebras are connected sums.^

#### Subject Area

Applied Mathematics|Mathematics|Theoretical Mathematics

#### Recommended Citation

Celikbas, Ela, "Prime ideals in two-dimensional Noetherian domains and fiber products and connected sums" (2012). *ETD collection for University of Nebraska - Lincoln*. AAI3523374.

https://digitalcommons.unl.edu/dissertations/AAI3523374