On polynomial and power series rings over a commutative ring

Automorphisms of Formal Power Series Rings over a Valuation Ring

The aim of this paper is to report on recent work on liftings of groups of au-tomorphisms of a formal power series ring over a eld k of characteristic p to characteristic 0, where they are realised as groups of automorphisms of a formal power series ring over a suitable valuation ring R dominating the Witt vectors W(k): We show that the lifting requirement for a group of automorphisms places se...

Reduced Gröbner Bases in Polynomial Rings over a Polynomial Ring

We define reduced Gröbner bases in polynomial rings over a polynomial ring and introduce an algorithm for computing them. There exist some algorithms for computing Gröbner bases in polynomial rings over a polynomial ring. However, we cannot obtain the reduced Gröbner bases by these algorithms. In this paper we propose a new notion of reduced Gröbner bases in polynomial rings over a polynomial r...

Standard bases in mixed power series and polynomial rings over rings

In this paper we study standard bases for submodules of a mixed power series and polynomial ring RJt1, . . . , tmK[x1, . . . , xn] s respectively of their localization with respect to a t-local monomial ordering for a certain class of noetherian rings R. The main steps are to prove the existence of a division with remainder generalizing and combining the division theorems of Grauert–Hironaka an...

Group Actions on Polynomial and Power Series Rings

Associated Graphs of Modules Over Commutative Rings

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. In this paper we introduce a new graph associated to modules over commutative rings. We study the relationship between the algebraic properties of modules and their associated graphs. A topological characterization for the completeness of the special subgraphs is presented. Also modules whose associated graph is complete...