相关论文: Countable and Full Exchange Rings

Let $R$ be a commutative Noetherian ring that is a smooth $\mathbb Z$-algebra. For each ideal $I$ of $R$ and integer $k$, we prove that the local cohomology module $H^k_I(R)$ has finitely many associated prime ideals. This settles a crucial…

For a commutative Noetherian local ring we define and study the class of modules having reducible complexity, a class containing all modules of finite complete intersection dimension. Various properties of this class of modules are given,…

We prove that every orientable infinite type surface without boundary and finite genus has a Riemann surface structure such that its modular group of quasiconformal homeomorphisms is countable.

We define, via Gorenstein homomorphisms, a class of local rings over which there exist non-trivial totally reflexive modules. We also provide a general construction of such rings, which indicates their abundance.

We introduce the class weakly nil clean rings, as rings R in which for every a\in R there exist an idempotent e and a nilpotent q such that a-e-q\in eRa. Every weakly nil clean ring is exchange. Weakly nil clean rings contain pi-regular…

Let K be a number field with euclidean ring of integers O. Let G be a finite-index torsion-free subgroup of Sp(2n, O). We exhibit a finite, geometrically defined spanning set of the top dimensional integral cohomology of G by generalizing…

We develop the basic properties of an essentially new closure operation on submodules, the \emph{liftable integral closure} of a submodule, including its relationships with the two prevailing notions of integral closure of submodules. We…

Given a minimal action $\alpha$ of a countable group on the Cantor set, we show that the alternating full group $\mathsf{A}(\alpha)$ is non-amenable if and only if the topological full group $\mathsf{F}(\alpha)$ is $C^*$-simple. This…

We obtain a characterization of left perfect rings via superstability of the class of flat left modules with pure embeddings. $\mathbf{Theorem.}$ For a ring $R$ the following are equivalent. - $R$ is left perfect. - The class of flat left…

We show that certain subrings of the cohomology of a finite p-group P may be realised as the images of restriction from suitable virtually free groups. We deduce that the cohomology of P is a finite module for any such subring. Examples…

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The aim of this paper is to introduce the notion of fully S-idempotent modules as a generalization of fully idempotent modules and investigate some…

We find a distributive (v, 0, 1)-semilattice S of size $ aleph\_1$ that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open…

A ring $R$ with center $C$ is said to be\textit{centrally essential} if the module $R_C$ is an essential extension of the module $C_C$. We describe centrally essential exterior algebras of finitely generated free modules over not necessary…

There are well-known constructions relating ring epimorphisms and tilting modules. The new notion of silting module provides a wider framework for studying this interplay. To every partial silting module we associate a ring epimorphism…

Let $\frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\V(\fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{\fa}_i(M)\neq 0$. We show that $M$ is $\fa$-cofinite if and…

The cohomology theory known as Tmf, for "topological modular forms," is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to…

A finitely generated module C over a commutative noetherian ring R is semidualizing if Hom_R(C,C) \cong R and Ext^i_R(C,C) = 0 for all i \geq 1. For certain local Cohen-Macaulay rings (R,m), we verify the equality of Hilbert-Samuel…

Let $R$ be a valuation ring and let $Q$ be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if $Q$ is maximal (respectively artinian). It is shown that each…

An E-ring is a unital ring R such that every endomorphism of the underlying abelian group R^+ is multiplication by some ring-element. The existence of almost-free E-rings of cardinality greater than 2^{aleph_0} is undecidable in ZFC. While…

Given a topological ring $R$, we study semitopological $R$-modules, construct their completions, Bohr and borno modifications. For every topological space $X$, we construct the free (semi)topological $R$-module over $X$ and prove that for a…