Standard bases in mixed power series and polynomial rings over rings

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...

Polynomial Rings over Pseudovaluation Rings

Let R be a ring. Let σ be an automorphism of R. We define a σ-divided ring and prove the following. (1) Let R be a commutative pseudovaluation ring such that x ∈ P for any P ∈ Spec(R[x,σ]) . Then R[x,σ] is also a pseudovaluation ring. (2) Let R be a σ-divided ring such that x ∈ P for any P ∈ Spec(R[x,σ]). Then R[x,σ] is also a σ-divided ring. Let now R be a commutative Noetherian Q-algebra (Q i...

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...

Projective modules over polynomial rings and dynamical Gröbner bases

Anti-archimedean Rings and Power Series Rings

We define an integral domain D to be anti-Archimedean if ⋂∞ n=1 a nD 6= 0 for each 0 6= a ∈ D. For example, a valuation domain or SFT Prüfer domain is anti-Archimedean if and only if it has no height-one prime ideals. A number of constructions and stability results for anti-Archimedean domains are given. We show that D is anti-Archimedean ⇔ D[[X1, . . .