Related papers: On the existence of parameterized noetherian rings

A ring $R$ is said to be centrally essential if for every its non-zero element $a$, there exist non-zero central elements $x$ and $y$ with $ax = y$. A ring $R$ is said to be completely centrally essential if all its factor rings are…

Let $k$ be a commutative Noetherian ring, and $k[S]$ the polynomial ring whose indeterminates are parameterized by elements in a set $S$. We show that $k[S]$ is Noetherian up to highly homogenous actions of groups. In particular, there is a…

Let K be a field of characteristic zero. Motivated by the conjecture that an enveloping algebra U(g) is Noetherian only if g is finite dimensional, we define the notion of weakly Noetherian Lie algebras. The main result, Theorem A, states…

Let $R$ be a commutative ring with identity. The ring $R\times R$ can be viewed as an extension of $R$ via the diagonal map $\Delta: R \hookrightarrow R\times R$, given by $\Delta(r) = (r, r)$ for all $r\in R$. It is shown that, for any $a,…

We introduce in this work, the class of commutative rings whose lattice of ideals forms an MTL-algebra which is not necessary a BL-algebra. The so-called class of rings will be named MTL-rings. We prove that a local commutative ring with…

In this paper, we define and study $S$-Noetherian lattices as a natural generalization of Noetherian rings. We prove that a ring $R$ is $S$-Noetherian if and only if its ideal lattice, $Id(R)$, is $S_L$-Noetherian. Furthermore, we establish…

In recent years, centrally essential rings have been intensively studied in ring theory. In particular, they find applications in homological algebra, group rings, and the structural theory of rings. The class of essentially central rings…

Given a noetherian local domain $R$ and a valuation $\nu$ of its field of fractions which is non negative on $R$, we derive some very general bounds on the growth of the number of distinct valuation ideals of $R$ corresponding to values…

Let $k$ be an uncountable algebraically closed field and let $A$ be a countably generated left Noetherian $k$-algebra. Then we show that $A \otimes_k K$ is left Noetherian for any field extension $K$ of $k$. We conclude that all subfields…

Let $\Gamma$ be a cancellation monoid and $R=\bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be a $\Gamma$-graded ring. It is shown that $R$ is graded left semihereditary if and only if $R$ is graded left coherent and every graded submodule of a…

The purpose of this article is to provide a new characterization of Cohen-Macaulay local rings. As a consequence we deduce that a local (Noetherian) ring $R$ is Gorenstein if and only if every parameter ideal of $R$ is irreducible.

We characterize which complete local (Noetherian) rings T containing the rationals are the completion of a countable excellent local ring S. We also discuss the possibilities for the map from the minimal prime ideals of T to the minimal…

We study relatively minimal subgroups in topological groups. We find, in particular, some natural relatively minimal subgroups in unipotent groups which are defined over "good" rings. By "good" rings we mean archimedean absolute valued (not…

In this paper we prove the following theorem. Let R be a prime Noetherian ring with krull dimension |R| = n where n is a positive integer. Let Q be the Goldie quotient ring of R. For a fixed positive integer m < n, let xm be the set of all…

Let $A$ be a nondegenerate dimer (or ghor) algebra on a torus, and let $Z$ be its center. Using cyclic contractions, we show the following are equivalent: $A$ is noetherian; $Z$ is noetherian; $A$ is a noncommutative crepant resolution;…

Let $K$ be a characteristic zero algebraic function field with a valuation $\nu$. Let $L$ be a finite extension of $K$ and $\omega$ be an extension of $\nu$ to $L$. We establish that the valuation ring $V_{\omega}$ of $\omega$ is…

We show that for a Noetherian ring $A$ that is $I$-adically complete for an ideal $I$, if $A/I$ admits a dualizing complex, so does $A$. This gives an alternative proof of the fact that a Noetherian complete local ring admits a dualizing…

We characterise Dedekind rings among not necessarily Noetherian domains by a property of their module homomorphisms. Our proof relies on a homological algebra argument.

Let $\Gamma$ be a countable group. A $\Gamma$-action on a compact abelian group $X$ by continuous automorphisms of $X$ is called Noetherian if the dual of $X$ is Noetherian as a ${\mathbb Z}(\Gamma)$-module. We prove that any Noetherian…

We define Dedekind semidomains as semirings in which each nonzero fractional ideal is invertible. Then we find some equivalent condition for semirings to being Dedekind. For example, we prove that a Noetherian semidomain is Dedekind if and…