Related papers: Characterization and examples of commutative isoar…

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, $M$ an arbitrary $R$-module and $N$ a finite $R$-module. We prove that \cite[Theorem 2.1]{Mel} and \cite[Proposition 3.3 (i)$\Leftrightarrow$(ii)]{B1} are true for any Serre…

A classical result of Micali asserts that a Noetherian local ring is regular if and only if the Rees algebra of its maximal ideal is defined by an ideal of linear forms. In this case, this defining ideal may be realized as a determinantal…

We count the number of submodules of an arbitrary module over a countable noetherian commutative ring. We give, along the way, a structural description of meager modules, which are defined as those that do not have the square of a simple…

There are several theorems describing the intricate relationship between flatness and associated primes over commutative Noetherian rings. However, associated primes are known to act badly over non-Noetherian rings, so one needs a suitable…

Let $B$ be a Noetherian normal local ring, and $G\subset\Aut(B)$ a cyclic group of local automorphisms of prime order. Let $A$ be the ring of $G$-invariants of $B$, assume that $A$ is Noetherian. We study the invariant morphism; in…

We prove some results on NIP integral domains, especially those that are Noetherian or have finite dp-rank. If $R$ is an NIP Noetherian domain that is not a field, then $R$ is a semilocal ring of Krull dimension 1, and the fraction field of…

Consider a Noetherian domain $R$ and a finite group $G \subseteq Gl_n(R)$. We prove that if the ring of invariants $R[x_1, \ldots, x_n]^G$ is a Cohen-Macaulay ring, then it is generated as an $R$-algebra by elements of degree at most…

Let $R$ be a commutative Noetherian ring and $\fa$ an ideal of $R$. We intend to establish the dual of two Faltings' Theorems for local homology modules of an Artinian module. As a consequence of this, we show that, if $A$ is an Artinian…

A torsion theoretical characterization of left Noetherian rings $R$ over which injective hulls of simple left modules are locally Artinian is given. Sufficient conditions for a left Noetherian ring to satisfy this finiteness condition are…

We investigate endoartinian modules, which satisfy the descending chain condition on endoimages, and establish new characterizations that unify classical and generalized chain conditions. Over commutative rings, endoartinianity coincides…

We show that the cosupport of a commutative noetherian ring is precisely the set of primes appearing in a minimal pure-injective resolution of the ring. As an application of this, we prove that every countable commutative noetherian ring…

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…

A graph is an instrument which is extensively utilized to model various problems in different fields. Up to date, many graphs have been developed to represent algebraic structures, particularly rings in order to study their properties. In…

This survey provides a comprehensive overview of the recent advancements in the theory of ``uniformly $S$''-algebraic structures in commutative ring theory. Originating from the classical concepts of Noetherian, coherent, von Neumann…

In this short note we study the links of certain prime ideals of a noetherian ring R. We first give the definition of a link krull symmetric noetherian ring R. We then prove theorem 9 that states that for any linked prime ideals P' and Q'…

Let R be a commutative Noetherian (not necessarily local) ring with identity and a be a proper ideal of R. We introduce a notion of a-relative system of parameters and characterize them by using the notion of cohomological dimension. Also,…

Noetherian dimer algebras form a prominent class of examples of noncommutative crepant resolutions (NCCRs). However, dimer algebras which are noetherian are quite rare, and we consider the question: how close are nonnoetherian homotopy…

Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$ and $M$ be an $R$-module. For a non-negative integer $n$ it is shown that, if the sets $\Ass_R(\Ext^{n} _{R}(R/I,M))$ and $\Supp_R(\Ext^{i}_{R}(R/I,H^{j}_{I,J} (M)))$ are…

We provide a natural criterion which implies equality of the finitistic test ideal and test ideal in local rings of prime characteristic. Most notably, we show that the criterion is met by every local weakly $F$-regular ring whose…

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…