Weak forms of the Krull-Schmidt theorem and Prüfer rings in distinguished constructions

This thesis is divided in two chapters. The first one concerns direct-sum decompositions in additive categories. It is well known that if a module admits a direct-sum decomposition into indecomposable modules with local endomorphism rings, then this decomposition is essentially unique, up to isomorphism and a permutation of the direct summands. However, there are situations in which direct-sum decompositions into indecomposable modules are not essentially unique. Among these cases, particularly interesting are those in which it is possible to find some kind of regularity: direct-sum decompositions can be described via two invariants up to two permutations. Such behaviour was firstly discovered for uniserial modules by A. Facchini in 1996, and it was subsequently investigated for several other classes of modules, such as cyclically presented modules over a local ring, couniformly presented modules and kernels of morphisms between indecomposable injective modules. In this thesis, we provide examples of additive categories in which direct-sum decompositions can be classified via finitely many invariants. It is worth noting that, in our constructions, we treat cases in which the number of invariants needed to describe finite direct-sum decompositions can be arbitrarily large. The second chapter is devoted to the study of Prufer (commutative) rings with zero-divisors. We investigate the so-called "Prufer-like conditions" in several constructions, most of them related to pullbacks. It is well known that fiber products provide a rich source of examples and counterexamples in Commutative Algebra, because of their ability of producing rings with certain predetermined properties. Our investigation moves from very natural settings, for example those of regular conductor squares, up to more technical constructions, such as bi-amalgamated algebras, introduced by Kabbaj, Louartiti and Tamekkante in 2017 as a generalization of that of amalgamated algebras. Our main results in the pullback framework cover several different situations studied up to now by Bakkaki and Mahdou, Boynton, Houston and Taylor. We also investigate Prufer ring from other points of view. We introduce the notion of regular morphism and we prove that if a ring R is the homomorphic image of a Prufer ring via a regular morphism, then R is Prufer. Finally, we turn our attention to the ideal-theory of pre-Prufer rings, proving a number of generalizations of some results of Boisen and Larsen.