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([special characters omitted][h, k]) over the integers [special characters omitted]. Roger Wiegand discovered an interesting axiom which holds for the affine line over [special characters omitted], that is, the partially ordered set of prime ideals in the polynomial ring [special characters omitted][x], but does not hold for Proj([special characters omitted][h, k]). In certain cases we are able to establish that a particular modification of his axiom holds in Proj([special characters omitted][h, k]). In addition we describe somewhat the behavior of the prime ideals of Proj([special characters omitted][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 NR so that, for every such indecomposable R-module M and every minimal prime ideal P of R, the dimension of MP, as a vector space over the field RP, is bounded by NR. 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 ≥ dimRP (MP) ≤ 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 ≤ dimRP(MP) ≤ 23; for every minimal prime ideal P of R