Let p be a fixed prime number. Throughout this paper, the symbols Z, Zp, Qp, and Cp denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers, and Z N ∪ {0}. Let νp be the normalized exponential valuation of Cp with |p|p p−νp p p−1 see 1–24 . Let UD Zp ...

The notion of a D-ring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the AxKochen-Ershov principle is proven for a theory of valued D-fields of residual characteristic zero. The model theory of differential and difference fields has been extensively studied (see for example [7, 3]) and valued fields have proven to be amenable t...

Let p be a fixed odd prime number. Throughout this paper Z, Q, Zp, Qp and Cp will respectively denote the ring of rational integers, the field of rational numbers, the ring p-adic rational integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp such that |p|p = p−vp(p) = p−1. If q ∈ Cp, we normally as...

Let p be prime. Let R be a discrete valuation ring of characteristic p with field of fractions K. Let C p denote the elementary abelian group of order p. In this paper we use Greither’s approach for classifying short exact sequences of finite R-group schemes to classify R-Hopf orders H in the group ring KC p , reproducing a result of Tossici. We then go further by providing an explicit descript...

Let us fix the usual notation • C is a geometrically connected smooth projective curve over a finite field Fq. • ∞ ∈ C is a closed point. • F = Fq(C) is the function field of C. • A = H(C − {∞},OC) is the ring of functions on C that are regular outside ∞. • Â = lim0 6=I⊂AA/I and Af = Â⊗A F . • F∞ is the completion of F at ∞ with valuation ring O∞. The ring A is discrete inside F∞, and the absol...

We prove that if $R$ is a commutative Noetherian ring, then every countably generated flat $R$-module quite flat, i.e., direct summand of transfinite extension localizations in countable multiplicative subsets. also show the spectrum cardinality less than $\kappa$, where $\kappa$ an uncountable regular cardinal, modules with generators. This provides alternative proof fact over ring spectrum, a...

We give a proposal for future development of the model theory of valued fields. We also summarize recent results on p-adic numbers. Let K be a valued field with a valuation map v : K → G ∪ {∞} to an ordered group 1 G; this is a map satisfying (i) v(x) = ∞ if and only if x = 0; (ii) v(xy) = v(x) + v(y) for all x, y ∈ K; (iii) v(x + y) ≥ min{v(x), v(y)} for all x, y ∈ K. We write R for the valuat...

It is proven that each indecomposable injective module over a valuation domain R is polyserial if and only if each maximal immediate extension R̂ of R is of finite rank over the completion R̃ of R in the R-topology. In this case, for each indecomposable injective module E, the following invariants are finite and equal: its Malcev rank, its Fleischer rank and its dual Goldie dimension. Similar res...