Using a local monomialization result of Knaf and Kuhlmann, we prove that the valuation ring an Abhyankar function field over perfect ground prime characteristic is Frobenius split. We show splitting sufficiently well-behaved center lifts to ring. also investigate properties valuations centered on arbitrary Noetherian domains characteristic. In contrast [arXiv:1507.06009], this paper emphasizes ...

For certain classes of Prüfer domains A, we study the completion Â,T ofA with respect to the supremum topology T = sup{Tw|w ∈ Ω}, where Ω is the family of nontrivial valuations on the quotient field which are nonnegative on A and Tw is a topology induced by a valuation w ∈ Ω. It is shown that the concepts ‘SFT Prüfer domain’ and ‘generalized Dedekind domain’ are the same. We show that if E is t...

Valuation theory is one of the main tools for studying higher level orders and the reduced theory of forms over fields, see, for example [BR]. In [MW], the theory of higher level orders and reduced forms was generalized to rings with many units and many of the results for fields carried over to this setting. While it seems desirable to extend these results further, the techniques used for rings...

Over any field an implicit linear difference equation one can reduce to the usual explicit one, which has infinitely many solutions ~ for each initial value. It is interesting consider over ring, because case of a ring significantly different from one. The previous results on equations rings mostly concern integers and low order equations. In present article high some other classes rings, parti...

A ring Λ is called a tiled order if Λ is a prime Noetherian semi-prefect semi-distributive ring with non-zero Jacobson radical. Any tiled order can be constructed from a (non-necessarily commutative) discrete valuation ring and an exponent matrix. The latter means a square integer matrix A = (αps), whose diagonal entries are equal to zero, and for all possible indices i, j, k, the following ine...

Contents 1. Introduction. 2. Heirs. 3. The invariance group of a cut. 4. Review of T-convex valuation rings. 5. The invariance valuation ring of a cut. 6. A method for producing cuts with prescribed signature. 7. Existentially closed extensions. 1. Introduction. Let M be a totally ordered set. A (Dedekind) cut p of M is a couple (p L , p R) of subsets p L , p R of M such that p L ∪ p R = M and ...

Recall that a regular local ring is a noetherian local ring with dimension equal to dimkm/m. A regular local ring of dimension one is precisely a discrete valuation ring. If V is a nosingular variety then t(V ) is a scheme with all local rings regular, so t(V ) is clearly regular in codimension one. In this section we will consider schemes satisfying the following condition: (∗) X is a noetheri...

Let $R= \oplus_{ \alpha \in G} R_{\alpha}$ be a commutative ring with unity graded by an arbitrary grading monoid $G$. For each positive integer, the notions of graded-n-coherent module and are introduced. In this paper many results generalized from $n$-coherent rings to graded-$n$-coherent rings. last section, we provide necessary sufficient conditions for trivial extension graded-valuation ri...