Related papers: Cohen's theorem in tensor triangular geometry

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 generalize Jacobson's notion of primitive ring to the setting of quantales. We show that every primitive ring gives rise to a primitive quantale of ideals. We then prove a density theorem for strongly primitive quantales. Furthermore, we…

We establish the primary decomposition and uniqueness of primary decomposition for k-ideals in commutative Noetherian semirings.

Let $R$ be a commutative ring with $1\neq 0$ and $n$ be a fixed positive integer. A proper ideal $I$ of $R$ is said to be an \textit{$n$-OA ideal} if whenever $a_1a_2\cdots a_{n+1}\in I$ for some nonunits $a_1,a_2,\ldots,a_{n+1}\in R$, then…

A conjecture raised in 1990 by C. Huneke predicts that, for a $d$-dimensional Noetherian local ring $R$, local cohomology modules of finitely generated $R$-modules have finitely many associated primes. Although counterexamples do exist, the…

This paper aims at developing a "local--global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the…

This paper investigates the Poisson geometry associated to a cluster algebra over the complex numbers, and its relationship to compatible torus actions. We show, under some assumptions, that each Noetherian cluster algebra has only finitely…

The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn's lemma, and thus pose a challenge from a computational point of view. Giving a…

Let $k$ be a field. We determine the ideals $I$ in a finitely generated graded $k$-algebra $A$, whose associated graded rings are isomorphic to $A$. Also we compute the graded local cohomologies of the Rees rings $A[I t]$ and give the…

Let $R$ be a commutative, local, Noetherian ring. In a past article, the first author developed a theory of $R$-algebras, termed seeds, that can be mapped to balanced big Cohen-Macaulay $R$-algebras. In prime characteristic $p$, seeds can…

We classify compactly generated co-t-structures on the derived category of a commutative noetherian ring. In order to accomplish that, we develop a theory for compactly generated Hom-orthogonal pairs (also known as torsion pairs in the…

We prove that a monomial ideal $I$ generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also…

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…

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…

This paper describes a new link between combinatorial number theory and geometry. The main result states that A is a finite set of relatively prime positive integers if and only if A = (K-K) \cap N, where K is a compact set of real numbers…

In this note we introduce and study basic properties of two types of modules over a commutative noetherian ring $R$ of positive prime characteristic. The first is the category of modules of finite $F$-type. These objects include reflexive…

Let R be an associative ring.In the paper we study n-generalized commutators of rings and prove that if R is a noncommutative prime ring and n > 2, then every nonzero n-generalized Lie ideal of R contains a nonzero ideal. Therefore, if R is…

Let $R$ be a normal Noetherian local domain of Krull dimension two. We examine intersections of rank one discrete valuation rings that birationally dominate $R$. We restrict to the class of prime divisors that dominate $R$ and show that if…

We introduce the notion of integrality of Grothendieck categories as a simultaneous generalization of the primeness of noncommutative noetherian rings and the integrality of locally noetherian schemes. Two different spaces associated to a…

A minor is principal means it is defined by the same row and column indices. Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of $X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let $\mathfrak P_t$…