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

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_P, as a vector space over the field R_P, is bounded by N_R. We show that, if R is such a ring of bounded representation type and if n ≥ 18, then every R-module M. such that n ≥ dimR_P (M_P) ≤ 2n − 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 2n − 13, that is, 18 ≤ dimR_P(M_P) ≤ 23; for every minimal prime ideal P of R.