Abelsche Galoiserweiterungen von R[X].
In this paper, we are concerned with G-rings. We generalize the Kaplansky’s theorem to rings with zero-divisors. Also, we assert that if is a ring extension such that for some regular element of , then is a G-ring if and only if so is . Also, we examine the transfer of the G-ring property to trivial ring extensions. Finally, we conclude the paper with illustrative examples discussing the utility and limits of our results.
A domain R is called an absolutely S-domain (for short, AS-domain) if each domain T such that R ⊆ T ⊆ qf(R) is an S-domain. We show that R is an AS-domain if and only if for each valuation overring V of R and each height one prime ideal q of V, the extension R/(q ∩ R) ⊆ V/q is algebraic. A Noetherian domain R is an AS-domain if and only if dim (R) ≤ 1. In Section 2, we study a class of R-subalgebras of R[X] which share many spectral properties with the polynomial ring R[X] and which we call pseudo-polynomial...
Let H be a Hopf algebra over a field k such that every finite-dimensional (left) H-module is semisimple. We give a counterpart of the first fundamental theorem of the classical invariant theory for locally finite, finitely generated (commutative) H-module algebras, and for local, complete H-module algebras. Also, we prove that if H acts on the k-algebra A = k[[X₁,...,Xₙ]] in such a way that the unique maximal ideal in A is invariant, then the algebra of invariants is a noetherian Cohen-Macaulay...
We connect the theorems of Rentschler [rR68] and Dixmier [jD68] on locally nilpotent derivations and automorphisms of the polynomial ring and of the Weyl algebra , both over a field of characteristic zero, by establishing the same type of results for the family of algebras where is an arbitrary polynomial in . In the second part of the paper we consider a field of prime characteristic and study comodule algebra structures on . We also compute the Makar-Limanov invariant of absolute constants...
In the article appeared in this same journal, vol. 33, 1 (1989) pp. 85-97, some statements in the proof of Example 3.4B got scrambled.
Let a be an ideal of a commutative ring A. There is a kind of duality between the left derived functors Uia of the a-adic completion functor, called local homology functors, and the local cohomology functors Hai.Some dual results are obtained for these Uia, and also inequalities involving both local homology and local cohomology when the ring A is noetherian or more generally when the Ua and Ha-global dimensions of A are finite.
It is explained that the following two problems are equivalent: (i) describing all affine rulings of any given weighted projective plane; (ii) describing all weighted-homogeneous locally nilpotent derivations of k[X,Y,Z]. Then the solution of (i) is sketched. (Outline of our joint work with Peter Russell.)
In my talk I am going to remind you what is the AK-invariant and give examples of its usefulness. I shall also discuss basic conjectures about this invariant and some positive and negative results related to these conjectures.