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# The projective line over the integers and decompositions of modules over one -dimensional rings

#### Abstract

This thesis is in two parts and concerns two topics in commutative algebra: (1) The projective line over the integers (Chapter 2), (2) Decompositions of modules over one-dimensional rings (Chapter 3). ^ In the first part of this thesis we give preliminary results towards a characterization of the underlying partially ordered set of the projective line Proj( Z [*h, k*]) over the integers Z . Roger Wiegand discovered an interesting axiom which holds for the affine line over Z , that is, the partially ordered set of prime ideals in the polynomial ring Z [*x*], but does not hold for Proj( Z [*h, k*]). In certain cases we are able to establish that a particular modification of his axiom holds in Proj( Z [*h, k*]). In addition we describe somewhat the behavior of the prime ideals of Proj( Z [*h, k*]) that are generated by prime integers. ^ The second part of the thesis contains an analysis of indecomposable finitely generated torsion-free modules over one-dimensional, reduced commutative Noe-therian rings with finite normalization. Such a ring *R* has *bounded representation type* if there exists a positive integer *N _{R}* so that, for every such indecomposable

*R*-module

*M*and every minimal prime ideal

*P*of

*R*, the dimension of

*M*, as a vector space over the field

_{P}*R*, is bounded by

_{P}*N*. We show that, if

_{R}*R*is such a ring of bounded representation type and if

*n*≥ 18, then every

*R*-module

*M*. such that

*n*≥ dim

*R*(

_{P}*M*) ≤ 2

_{P}*n*− 14 for every minimal prime ideal

*P*of

*R*; must decompose non-trivially. We also give an example of a semilocal ring-order

*R*of bounded representation type and an indecomposable module

*M*such that all the relevant dimensions are between

*n*= 18 and 2

*n*− 13, that is, 18 ≤ dim

*R*(

_{P}*M*) ≤ 23; for every minimal prime ideal

_{P}*P*of

*R*. ^

#### Subject Area

Mathematics

#### Recommended Citation

Arnavut, Meral, "The projective line over the integers and decompositions of modules over one -dimensional rings" (2002). *ETD collection for University of Nebraska - Lincoln*. AAI3059938.

http://digitalcommons.unl.edu/dissertations/AAI3059938