On locally quasi A∗ algebras in codimension-one over a Noetherian normal domain

AbstractLet R be Noetherian normal domain. We shall call an R-algebra A quasi A∗ if A=R[X,(aX+b)−1] where X∈A is a transcendental element over R, a∈R∖0, b∈R and (a,b)R=R. In this paper we shall describe a general structure for any faithfully flat R-algebra A which is locally quasi A∗ in codimension-one over R. We shall also investigate minimal sufficient conditions for such an algebra to be finitely generated

Strongly residual coordinates over A[x]

For a domain A of characteristic zero, a polynomial f over A[x] is called a strongly residual coordinate if f becomes a coordinate (over A) upon going modulo x, and f becomes a coordinate upon inverting x. We study the question of when a strongly residual coordinate is a coordinate, a question closely related to the Dolgachev-Weisfeiler conjecture. It is known that all strongly residual coordinates are coordinates for n=2 . We show that a large class of strongly residual coordinates that are generated by elementaries upon inverting x are in fact coordinates for arbitrary n, with a stronger result in the n=3 case. As an application, we show that all Venereau-type polynomials are 1-stable coordinates.Comment: 15 pages. Some minor clarifications and notational improvements from the first versio

Zero-divisor graphs of nilpotent-free semigroups

We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an \emph{Armendariz map} between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtaining surprisingly strong structure theorems relating ring-theoretic and topological properties to graph-theoretic invariants of the corresponding graphs.Comment: Expanded first paragraph in section 6. To appear in J. Algebraic Combin. 22 page

The structure of a Laurent polynomial fibration in n variables

AbstractBass, Connell and Wright have proved that any finitely presented locally polynomial algebra in n variables over an integral domain R is isomorphic to the symmetric algebra of a finitely generated projective R-module of rank n. In this paper we prove a corresponding structure theorem for a ring A which is a locally Laurent polynomial algebra in n variables over an integral domain R, viz., we show that A is isomorphic to an R-algebra of the form (SymR(Q))[I−1], where Q is a direct sum of n finitely generated projective R-modules of rank one and I is a suitable invertible ideal of the symmetric algebra SymR(Q). Further, we show that any faithfully flat algebra over a Noetherian normal domain R, whose generic and codimension-one fibres are Laurent polynomial algebras in n variables, is a locally Laurent polynomial algebra in n variables over R