Related papers: Sequence-regular commutative DG-rings

Let $R$ be a complete regular local ring with an algebraically closed residue field and let $A$ be a Noetherian $R$-subalgebra of the polynomial ring $R[X]$. It has been shown in \cite{DO2} that if $\dim R=1$, then $A$ is necessarily…

We introduce a notion of generalized homogeneous derivations on graded rings as a natural extension of the homogeneous derivations defined by Kanunnikov. We then define gr-generalized derivations, which preserve the degrees of homogeneous…

We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective…

In this article we extend a previous definition of Castelnuovo-Mumford regularity for modules over an algebra graded by a finitely generated abelian group. Our notion of regularity is based on Maclagan and Smith's definition, and is…

Local log-regular rings are a certain class of Cohen-Macaulay local rings that are treated in logarithmic geometry. Our paper aims to provide purely commutative ring theoretic proof of some ring-theoretic properties of local log-regular…

Let $k$ be an arbitrary field, $\Lambda$ be a $k$-algebra and $V$ be a $\Lambda$-module. When it exists, the universal deformation ring $R(\Lambda,V)$ of $V$ is a $k$-algebra whose local homomorphisms to $R$ parametrize the lifts of $V$ up…

Let $(A,\mathfrak{m})$ be an excellent Henselian Cohen-Macaulay local ring of finite representation type. If the AR-quiver of $A$ is known then by a result of Auslander and Reiten one can explicity compute $G(A)$ the Grothendieck group of…

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally,…

Let $(A,\mathfrak{m})$ be a complete equicharacteristic Noetherian domain of dimension $d + 1 \geq 2$. Assume $k = A/\mathfrak{m}$ has characteristic zero and that $A$ is not a regular local ring. Let $Sing(A)$ the singular locus of $A$ be…

This paper gives a necessary and sufficient condition for Gorensteinness in Rees algebras of the $d$-th power of parameter ideals in certain Noetherian local rings of dimension $d\ge 2$. The main result of this paper produces many…

Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB-rings. These constitute a considerable enlargement of the class of rings with stable rank one (B-rings), and…

Let $R$ be a Noetherian local ring. We prove that $R$ is regular of dimension at most four if, and only if, every prime ideal, defining a Gorenstein quotient ring, is syzygetic. We deduce a characterization of these rings in terms of the…

In this paper, we study the properties of noetherian rings with uniform annihilators. It turns out that all these rings should be universally catenary and locally equidimensional. We will give a necessary and sufficient condition for these…

Let $R$ be a commutative ring with identity, $S\subseteq R$ be a multiplicative set and $J$ be an ideal of $R$. In this paper, we introduce the concept of $S$-$J$-Noetherian rings, which generalizes both $J$-Noetherian rings and…

Let $(R,m)$ be a Noetherian local ring and $I$ an ideal with finite projective dimension. If $R/I$ satisfies some property $\mathcal{P}$, it is natural to ask whether $R$ would also satisfy this property $\mathcal{P}$. This is called the…

Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ be an ideal of $R$. Suppose $M$ is a finitely generated $R$-module and $N$ is an Artinian $R$-module. We define the concept of filter coregular sequence to determine the infimum of…

The rank of a ring $R$ is the supremum of minimal cardinalities of generating sets of $I$, among all ideals $I$ in $R$. In this paper, we obtain a characterization of Noetherian rings $R$ whose rank is not equal to the supremum of ranks of…

Let $(R,\mathfrak{m},\mathbb{k})$ be an equicharacteristic one-dimensional complete local domain over an algebraically closed field $\mathbb{k}$ of characteristic 0. R. Berger conjectured that R is regular if and only if the universally…

We introduce quasi-Gorenstein morphisms of commutative local dg-algebras and use a Gorenstein version of the virtually small property to characterize them, a result which is new even for homomorphisms of local rings. In a different…

Let $D$ be an integrally closed local Noetherian domain of Krull dimension 2, and let $f$ be a nonzero element of $D$ such that $fD$ has prime radical. We consider when an integrally closed ring $H$ between $D$ and $D_f$ is determined…