Gorenstein weak global dimension is symmetric

We define the presented dimensions for modules and rings to measure how far away a module is from having an infinite finite presentation and develop ways to compute the projective dimension of a module with a finite presented dimension and the right global dimension of a ring. We also make a comparison of the right global dimension, the weak global dimension, and the presented dimension and divide rings into four classes according to these dimensions.

Global dimensions of power series rings

In this paper we characterize the relative Gorenstein weak global dimension of the Gorenstein $$\mathcal {B}$$ B -flat R-modules and projectively coresolved Gorenstein $$\mathcal {B}$$ B -flat R-modules recently studied by S. Estrada, A. Iacob, and M. A. Pérez, which are a relativisation of the ones introduced by J. Šaroch and J. Št’ovíchěk. As application we prove that the weak global dimension with respect to the Gorenstein $$\textrm{FP}_n$$ FP n -flat R-modules is finite over a Gorenstein n-coherent ring R and in this case coincides with the flat dimension of the right $$\textrm{FP}_n$$ FP n -injective R-modules. This result extends the known for Gorenstein flat modules over Iwanaga-Gorenstein and Ding-Chen rings. We also show that there is a close relationship between the relative global dimension of the Gorenstein $$\textrm{FP}_n$$ FP n -projectives and the Gorenstein weak global dimension respect to the class of Gorenstein $$\textrm{FP}_n$$ FP n -flat R-modules. We also get an hereditary and complete cotorsion triple and consequently a balanced pair.

Let f : A! B be a ring homomorphism and J be an ideal of B. In this paper, we give a characterization of valuation, weak global dimension and semihereditary properties under a certain ring-theoretic construction called the amalgamation of A with B along J with respect to f (denoted by A ./ f J); introduced and studied by D’Anna, Finocchiaro and Fontana in 2009. Our aim is to generated new classes of commutative rings satisfying theses properties.

We prove that ifR is a commutative ring such that each localization at a prime ideal has finite weak global dimension then everyR-module has a flat envelope if and only ifR is coherent and has weak global dimension less than or equal to two.

Coherent H∞(D)-Modules in Control Theory

Global dimension in serial rings

Recently, there have been many results which show that the global dimension of certain rings can be computed using a proper subclass of the cyclic modules, e.g., the simple modules. In this paper we view calculating global dimensions in this fashion as a property of a ring and show that this is a property which transfers to the ring's idealizer and subidealizer ring.

Global and krull dimension of schmidt differentialoperator rings

In 2002, Anderson and Dumitrescu introduced [Formula: see text]-Noetherian rings, a significant advancement in ring theory. Later, in 2018, Bennis and El Hajoui expanded on this by introducing two new classes of rings that generalize the concept of coherent rings: [Formula: see text]-coherent rings, where every finitely generated ideal is [Formula: see text]-finitely presented, and [Formula: see text]-[Formula: see text]-coherent rings, where every [Formula: see text]-finite ideal is [Formula: see text]-finitely presented. This paper builds on these concepts to introduce the [Formula: see text]-[Formula: see text]-(weak) global dimension for rings with a multiplicative subset [Formula: see text], offering a new perspective on ring structures. We also explore [Formula: see text]-[Formula: see text]-variants of specific ring types, including semisimple, von Neumann, and (semi)hereditary rings. Finally, we extend the generalization of [Formula: see text]-rings and [Formula: see text]-coherent rings, originally introduced by Costa in 1994, to rings with a multiplicative subset [Formula: see text], deepening the understanding of ring theory.

(Non)Vanishing results on local cohomology of valuation rings

Projective, injective, and global dimensions.The projective and injective dimensions of a moduleMwill be denoted by p.dim(M) and inj dim(M), respectively. Theright global dimensionof a ringR, denoted r.gl.dim(R), is the supremum of the projective dimensions of the right R-modules. Theleft global dimensionis defined analogously. Recall that the right and left global dimensions of a noetherian ring coincide[190,Corollary 9.23]. Hence, we just write gl.dim(R) for this dimension. It is also known that if inj.dim(RR) and inj.dim(RR) are both finite, then these numbers coincide[216,Lemma A]. In this case, we just write inj.dim(R) for the common value.

This paper is concerned with finding the global homo-

Let $R$ be a Cohen-Macaulay ring and $M$ a maximal Cohen-Macaulay $R$-module. Inspired by recent striking work by Iyama, Burban-Iyama-Keller-Reiten and van den Bergh we study the question of when the endomorphism ring of $M$ has finite global dimension via certain conditions about vanishing of Ext modules. We are able to strengthen certain results by Iyama on connections between a higher dimension version of Auslander correspondence and existence of non-commutative crepant resolutions. We also recover and extend to positive characteristics a recent Theorem by Burban-Iyama-Keller-Reiten on cluster-tilting objects in the category of maximal Cohen-Macaulay modules over reduced $1$-dimensional hypersurfaces.

A module is called distributively generated if it is the sum of its distributive submodules. A ring with a distributive lattice of ideals is called an arithmetical ring. A ring is said to be right invariant if all its right ideals are ideals. We write w. gl.dim(A) 1 if all submodules of flat right (left) A-modules are flat. If A is commutative, then w. gl.dim(A) 1 ⇐⇒ A is a distributive semiprime ring [1].