Based on the work of Hahn, Baer, Ostrowski, Krull, Kaplansky and the Artin-Schreier theory, and stimulated by the paper [L] of S. Lang in 1953, the theory of real places and convex valuations has witnessed a remarkable development and has become a basic tool in the theory of ordered fields and real algebraic geometry. Surveys on this development can be found in [LAM] and [PC]. In this paper, we...

Classically, Gröbner bases are computed by first prescribing a set monomial order. Moss Sweedler suggested an alternative and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on k(x, y) that are suitable for this framework. For these valuations, we compute ν(k[x, y]∗) and use this ...

Let A be a commutative ring and E a non-zero A-module. Necessary and sufficient conditions are given for the trivial ring extension R of A by E to be either arithmetical or Gaussian. The possibility for R to be Bézout is also studied, but a response is only given in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected. Trivial ring extensions which are fqp-rings are cha...

Let $\mathcal R $ be a finite valuation ring of order $q^r$ with $q$ power an odd prime number, and let A \subset \mathcal $. We improve recent result due to Yazici (2018) on sum-product type problem. More precisely, we prove that

It is proved that EJ is injective if E is an injective module over a valuation ring R, for each prime ideal J 6= Z. Moreover, if E or Z is flat, then EZ is injective too. It follows that localizations of injective modules over h-local Prüfer domains are injective too. If S is a multiplicative subset of a noetherian ring R, it is well known that SE is injective for each injective R-module E. The...

For an o-minimal expansion R of a real closed field and a set V of Th(R)-convex valuation rings, we construct a “pseudo completion” with respect to V . This is an elementary extension S of R generated by all completions of all the residue fields of the V ∈ V , when these completions are embedded into a big elementary extension of R. It is shown that S does not depend on the various embeddings u...

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We give an explicit construction of the antiequivalence of the category of finite flat commutative group schemes of period 2 defined over a valuation ring of a 2-adic field with algebraically closed residue field. This result extends the earlier author’s approach to group schemes of period p > 2 from Proceedings LMS, 101, 2010, 207-259.

We show that standard arguments for deformations based on dimension counts can also be applied over a (not necessarily Noetherian) valuation ring A of rank 1. Key intermediate results are a principal ideal theorem for schemes of finite type over A, and a theorem on subadditivity of intersection codimension for schemes smooth over A.