Power series over Noetherian domains, Nagata rings, and Kronecker function rings

A Generalization of Kronecker Function Rings and Nagata Rings

Let D be an integral domain with quotient field K. The Nagata ring D(X) and the Kronecker function ring Kr(D) are both subrings of the field of rational functions K(X) containing as a subring the ring D[X] of polynomials in the variable X. Both of these function rings have been extensively studied and generalized. The principal interest in these two extensions ofD lies in the reflection of vari...

Nagata Rings, Kronecker Function Rings and Related Semistar Operations

In 1994, Matsuda and Okabe introduced the notion of semistar operation. This concept extends the classical concept of star operation (cf. for instance, Gilmer’s book [20]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer and P. Lorenzen from 1930’s. In [17] and [18] the current authors investigated properties of the Kronecker functi...

Noetherian Skew Inverse Power Series Rings

We study skew inverse power series extensions R[[y−1; τ, δ]], where R is a noetherian ring equipped with an automorphism τ and a τ -derivation δ. We find that these extensions share many of the well known features of commutative power series rings. As an application of our analysis, we see that the iterated skew inverse power series rings corresponding to nth Weyl algebras are complete, local, ...

An historical overview of Kronecker function rings, Nagata rings, and related star and semistar operations

Graded Integral Domains and Nagata Rings , Ii

Let D be an integral domain with quotient field K, X be an indeterminate over D, K[X] be the polynomial ring over K, and R = {f ∈ K[X] | f(0) ∈ D}; so R is a subring of K[X] containing D[X]. For f = a0 + a1X + · · ·+ anX ∈ R, let C(f) be the ideal of R generated by a0, a1X, . . . , anX n and N(H) = {g ∈ R | C(g)v = R}. In this paper, we study two rings RN(H) and Kr(R, v) = { fg | f, g ∈ R, g 6=...