Related papers: Power Series Rings over Zero-Dimensional Rings

In this note, we study the generalized fraction properties and power series properties of $\mathcal{S}$-Noetherian rings. Actually, we investigate two questions proposed in [A. Dabbabi, A. Benhissi, Generalization of the $S$-Noetherian…

We study rings over which an analogue of the Weierstrass preparation theorem holds for power series. We show that a commutative ring $R$ admits a factorization of every power series in $R[[x]]$ as the product of a polynomial and a unit if…

A ring R shall be called F-noetherian if every finite subset of R is contained in a (left and right) noetherian subring of R . For example, every commutative ring is tightly F-noetherian in the sense that every finite subset of R generates…

Let $R$ be a Noetherian ring. We prove that $R$ has global dimension at most two if, and only if, every prime ideal of $R$ is of linear type. Similarly, we show that $R$ has global dimension at most three if, and only if, every prime ideal…

Let $k$ be a perfect field of characteristic $p>0$, $k(t)_{per}$ the perfect closure of $k(t)$ and $A$ a $k$-algebra. We characterize whether the ring $A\otimes_k k(t)_{per}$ is noetherian or not. As a consequence, we prove that the ring…

Let R* be an ideal-adic completion of a Noetherian integral domain R and let L be a subfield of the total quotient ring of R* such that L contains R. Let A denote the intersection of L with R*. The integral domain A sometimes inherits nice…

The strong global dimension of a ring is the supremum of the length of perfect complexes that are indecomposable in the derived category. In this note we characterize the noetherian commutative rings that have finite strong global…

We show that the Rouquier dimension of the category of perfect complexes over a regular ring is precisely the Krull dimension of the ring. Previously, it was known that the Krull dimension is an upper bound, the lower bound however was not…

For a finitely generated algebra over a field, the transcendence degree is known to be equal to the Krull dimension. The aim of this paper is to generalize this result to algebras over rings. A new definition of the transcendence degree of…

We introduce a naive notion of a system of parameters for a homologically finite complex over a commutative noetherian local ring, and compare it to the system of parameters defined by Christensen. We show that these notions differ in…

Unlike the Gorenstein projective and injective dimensions, the majority of results on the Gorenstein flat dimension have been established only over Noetherian (or coherent) rings. Naturally, one would like to generalize these results to any…

We show that a formal power series ring $A[[X]]$ over a noetherian ring $A$ is not a projective module unless $A$ is artinian. However, if $(A,{\mathfrak m})$ is local, then $A[[X]]$ behaves like a projective module in the sense that…

We define and study two generalizations of the Krull dimension for rings, which can assume cardinal number values of arbitrary size. The first, which we call the "cardinal Krull dimension," is the supremum of the cardinalities of chains of…

The theory of border bases for zero-dimensional ideals has attracted several researchers in symbolic computation due to their numerical stability and mathematical elegance. As shown in (Francis & Dukkipati, J. Symb. Comp., 2014), one can…

We study rings which have Noetherian cohomology under the action of a ring of cohomology operators. The main result is a criterion for a complex of modules over such a ring to have finite injective dimension. This criterion generalizes, by…

In this article, a new and natural topology on the prime spectrum is established which behaves completely as the dual of the Zariski topology. It is called the flat topology. The basic and also some sophisticated properties of the flat…

K. R. Nagarajan constructed an example of a formal power series ring of dimension two, over a field of characteristic two, with the action of a cyclic group of order two, such that the ring of invariants is not noetherian. We point out how…

In 2007, Y. Shimoda, in connection with a long-standing question of J. Sally, asked whether a Noetherian local ring, such that all its prime ideals different from the maximal ideal are complete intersections, has Krull dimension at most…

Noetherian rings have played a fundamental role in commutative algebra, algebraic number theory, and algebraic geometry. Along with their dual, Artinian rings, they have many generalizations, including the notions of isonoetherian and…

We introduce the notion of Krull super-dimension of supermodules over certain super-commutative Noetherian super-rings. We investigate how this notion relates to the notion of odd regular sequence introduced by T.Schmitt and how it behaves…