Let G = Dp be the dihedral group of order 2p, where p is an odd prime. Let k an algebraically closed field of characteristic p. We show that any action of G on the ring k[[y]] can be lifted to an action on R[[y]], where R is some complete discrete valuation ring with residue field k and fraction field of characteristic 0. 2000 Mathematics subject Classification. Primary 14H37. Secondary: 11G20,...

SV-rings are commutative rings whose factor rings modulo prime ideals are valuation rings. SV-rings occur most naturally in connection with partially ordered rings (= porings) and have been studied only in this context so far. The present note first develops the theory of SV-rings systematically, without assuming the presence of a partial order. Particular attention is paid to the question of a...

We show that every injective homological ring epimorphism f : R → S where SR has flat dimension at most one gives rise to a 1cotilting R-module and we give sufficient conditions under which the converse holds true. Specializing to the case of a valuation domain R, we illustrate a bijective correspondence between equivalence classes of injective homological ring epimorphisms originating in R and...

Ayache has recently proved that if R is an integrally closed domain such that each overring of R is treed, then R is a locally pseudo-valuation domain. We investigate the extent to which the analogue (in which one concludes that R is a locally pseudo-valuation ring) holds if R is generalized to a commutative ring. A positive result is obtained if R is an idealization D(+)K where D is an integra...

We define and study a notion of ring of formal power series with exponents in a cyclically ordered group. Such a ring is a quotient of various subrings of classical formal power series rings. It carries a two variable valuation function. In the particular case where the cyclically ordered group is actually totally ordered, our notion of formal power series is equivalent to the classical one in ...

We present a set of generators of the full annihilator ideal for the Witt ring of an arbitrary field of characteristic unequal to two satisfying a non-vanishing condition on the powers of the fundamental ideal in the torsion part of the Witt ring. This settles a conjecture of Ongenae and Van Geel. This result could only be proved by first obtaining a new lower bound on the 2-adic valuation of S...

For a field F of characteristic different from 2, containing a square root of -1, endowed with an F-compatible valuation v such that the residue field has at most two square classes, we use a combinatorial analogue of the Witt ring of F to prove that an anisotropic quadratic form over F with even dimension d, trivial discriminant and Hasse-Witt invariant can be written in the Witt ring as the s...

Given a well-ordered semi-group Γ with a minimal system of generators of ordinal type at most ωn 1 and of rational rank r, which satisfies a positivity and increasing condition, we construct a zero-dimensional valuation centered on the ring of polynomials with r variables such that the semigroup of the values of the polynomial ring is equal to Γ. The construction uses a generalization of Favre ...

We compute explicitly the normal zeta functions of the Heisenberg groups H(R), where R is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form H(OK), where OK is the ring of integers of an arbitrary number field K, at the rational primes which are non-split in K. We show that these local z...

It is shown that every dp-minimal integral domain R a local ring and for non-maximal prime ideal ℘ of R, the localization R℘ valuation ℘R℘ = ℘. Furthermore, if only its residue field infinite or finite maximal principal.