… R is a valuation ring we will assume it has rank 1, and we will usually assume that its value
group is a proper, dense subgroup of R . In section 2, we will explore some valuation theory, …

… V is trivial if it is trivial over K. This result is predicted by the … brought by general nondiscrete
valuation rings. By using flasque … on formal power series over valuation rings and patching of …

… We also show in Chapter 3 that one can use the t-operation to uniquely characterize the (Manis)
valuation rings: a ring R is a valuation ring if and only if R is a PVMR that has at most …

… Let C be a smooth projective curve over a discretely valued field K, defined by an affine
equation fx;y/ D 0. We construct a model of C over the ring of integers of K using a toroidal …

… Rational valuations of N0[X] are special cases of Puiseux monoids, … rings [11] and upper
triangular matrices over information semialgebras [3]. We put emphasis on algebraic valuations …

… subset associated to a flat ring epimorphism. In case the underlying ring is locally factorial
or of Krull dimension one, we show that all flat ring epimorphisms are universal localizations. …

… noetherian ring is regular exactly when it admits a faithfully flat map to a perfectoid ring. This
… of finitely generated modules over noetherian rings via maps to perfectoid rings. We also …

… module over a Gorenstein local ring. In Chapter 4 we introduce flat complexes. We prove
that over a commutative noetherian ring with finite Krull dimension any complex has a …

… A ring is obviously a semiring with an additive inverse for each element. A module over a ring
is a generalisation of vector space over … ring (with identity) and the elements of the ring and …

… of modules. We show two torsion pairs with respect to the FP n -injective modules and the
FP n -flat modules over n-hereditary rings. We also provide an example of a Bézout ring which …