Let $(K,v)$ be a valued field. We review some results of MacLane and Vaqui\'e on extensions $v$ to valuations the polynomial ring $K[x]$. introduce certain MacLane-Vaqui\'e chains residually transcendental valuations, we prove that every valuation $\mu$ $K[x]$ is limit finite or countably infinite chain. This chain underlying essentially unique contains arithmetic data yielding an explicit desc...

Given a ring R, we introduce the notion of a generalized Laurent polynomial ring over R. This class includes the generalized Weyl algebras. We show that these rings inherit many properties from the ground ring R. This construction is then used to create two new families of quadratic global dimension four Artin-Schelter regular algebras. We show that in most cases the second family has a finite ...

We prove the following form of Dirichlet’s theorem for polynomial rings in one indeterminate over a pseudo algebraically closed field F . For all relatively prime polynomials a(X), b(X) ∈ F [X] and for every sufficiently large integer n there exist infinitely many polynomials c(X) ∈ F [X] such that a(X) + b(X)c(X) is irreducible of degree n, provided that F has a separable extension of degree n.

Let Fq be a finite field of characteristic p, K a field containing it, and R = K[X1, . . . , Xn] a polynomial ring in n variables. The general linear group GLn(Fq) has natural action on R by degree preserving ring automorphisms. L. E. Dickson showed that the subring of elements which are fixed by this group action is a polynomial ring, [Di], though for an arbitrary subgroupG of GLn(Fq), the str...

Quillen's solution of Serre's problem is extended to Laurent polynomial rings. An example is given of a A[T, r~']-module P which is not extended even though A is regular and Pm is extended for all maximal ideals m of A. The object of this note is to present several comments and examples related to some problems suggested by Quillen's recent solution of Serre's problem [7]. It is an immediate co...

We prove a Dirichlet theorem for polynomial rings: Let F be a pseudo algebraically closed field (i.e., each nonempty variety defined over F has an F-rational point). Then for all relatively prime polynomials a(X), b(X) ∈ F [X] and for every sufficiently large integer n there exist infinitely many polynomials c(X) ∈ F [X] such that a(X) + b(X)c(X) is irreducible of degree n, provided that F has ...

CARL FAITH All rings considered are commutative with unit. A ring R is SISI (in Vámos' terminology) if every subdirectly irreducible factor ring R/I is self-injective . SISI rings include Noetherian rings, Morita rings, and almost maximal valuation rings ([Vil) . In [F3] we raised the question of whether a polynomial ring R[-1 over a SISI ring R is again SISI . In this paper we show this is not...

A finitely generated Λ = Z[t, t]-module without Z-torsion is determined by a pair of sub-lattices of Λ. Their indices are the absolute values of the leading and trailing coefficients of the order of the module. This description has applications in knot theory. MSC 2010: Primary 13E05, 57M25.