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Let $R$ be a Noetherian ring. We prove that $R$ has global dimension at most two if, and only if, every prime ideal of $R$ is of linear type. Similarly, we show that $R$ has global dimension at most three if, and only if, every prime ideal…

We introduce a naive notion of a system of parameters for a homologically finite complex over a commutative noetherian local ring, and compare it to the system of parameters defined by Christensen. We show that these notions differ in…

We develop a new technique for studying monomial ideals in the standard polynomial rings $A[X_1,\ldots,X_d]$ where $A$ is a commutative ring with identity. The main idea is to consider induced ideals in the semigroup ring…

In 2007, Y. Shimoda, in connection with a long-standing question of J. Sally, asked whether a Noetherian local ring, such that all its prime ideals different from the maximal ideal are complete intersections, has Krull dimension at most…

Let $A$ be a commutative arithmetical ring. The ring $A$ has Krull dimension if and only if every factor ring of $A$ is finite-dimensional and does not have idempotent proper essential ideals. The study is supported by Russian Science…

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…

Let (R,m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If G_C-dimension of M/IM is finite for all ideals I generated by an R-regular sequence of length…

Let $P$ be a finitely generated ideal of a commutative ring $R$. Krull's Principal Ideal Theorem states that if $R$ is Noetherian and $P$ is minimal over a principal ideal of $R$, then $P$ has height at most one. Straightforward examples…

The injective spectrum is a topological space associated to a ring $R$, which agrees with the Zariski spectrum when $R$ is commutative noetherian. We consider injective spectra of right noetherian rings (and locally noetherian Grothendieck…

This paper explores the dimension theory of non-Noetherian graded rings by introducing the class of Hilbert-Serre rings. We generalize Krull's dimension theorem and Smoke's dimension theorem by establishing the fundamental inequalities…

We introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence. Our main result shows that the Krull dimension of a Noetherian ring is equal to the…

In the derived category of the category of modules over a commutative Noetherian ring $R$, we define, for an ideal $\fa$ of $R$, two different types of cohomological dimensions of a complex $X$ in a certain subcategory of the derived…

In this paper, we study Noetherian local rings $R$ having a finite number of trace ideals. We proved that such rings are of dimension at most two. Furthermore, if the integral closure of $R/H$, where $H$ is the zeroth local cohomology, is…

Let $R$ be a commutative ring, $Y\subseteq \mathrm{Spec}(R)$ and $ h_Y(S)=\{P\in Y:S\subseteq P \}$, for every $S\subseteq R$. An ideal $I$ is said to be an $\mathcal{H}_Y$-ideal whenever it follows from $h_Y(a)\subseteq h_Y(b)$ and $a\in…

In this paper, we introduce and study the $q$-Krull dimension of a commutative ring via its $q$-operation. A new characterization of $\tau_q$-von Neumann regular rings is obtained, and some properties of rings $q$-Krull dimension 0 are…

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 introduce the notion of Krull super-dimension of a super-commutative super-ring. This notion is used to describe regular super-rings and calculate Krull super-dimensions of completions of super-rings. Moreover, we use this notion to…

We prove some results on the structure of ind-pro completions of Noetherian rings along flags of prime ideals. In particular, we compute the Krull dimension and deduce the criterion on semilocality in the case of essentially of finite type…

Let R denote a commutative Noetherian ring and let I be an ideal of R such that H_i^I(R) = 0, for all integers i greater than or equal to 2. In this paper we shall prove some results concerning the homological properties of I.

Over a commutative noetherian ring $R$ of finite Krull dimension, we show that every complex of flat cotorsion $R$-modules decomposes as a direct sum of a minimal complex and a contractible complex. Moreover, we define the notion of a…