Related papers: Injective generation for graded rings

We found a necessary and sufficient condition for the existence of the tensor product of modules over a vertex algebra. We defined the notion of vertex bilinear map and we provide two algebraic construction of the tensor product, where one…

For a commutative noetherian ring $R$, we classify all the hereditary cotorsion pairs cogenerated by pure-injective modules of finite injective dimension. The classification is done in terms of integer-valued functions on the spectrum of…

This is the first part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. This theory generalizes the tensor category theory for…

We study symmetry-enriched topological order in two-dimensional tensor network states by using graded matrix product operator algebras to represent symmetry induced domain walls. A close connection to the theory of graded unitary fusion…

We consider the finite generation property for cohomology algebra of pointed finite tensor categories via de-equivariantization and exact sequence of finite tensor categories. As a result, we prove that all coradically graded pointed finite…

Invariable generation is a topic that has predominantly been studied for finite groups. In 2014, Kantor, Lubotzky, and Shalev produced extensive tools for investigating invariable generation for infinite groups. Since their paper, various…

Motivated by the Bass conjecture, we study finitely generated modules of finite injective dimension and the additional constraints they impose on the ambient ring. Beyond the Cohen--Macaulay property, the existence of such modules forces…

We develop the Morita theory of fusion 2-categories. In order to do so, we begin by proving that the relative tensor product of modules over a separable algebra in a fusion 2-category exists. We use this result to construct the Morita…

The rapid advancements in large language models and generative artificial intelligence (AI) capabilities are making their broad application in the high-stakes testing context more likely. Use of generative AI in the scoring of constructed…

We present the Insertion Transformer, an iterative, partially autoregressive model for sequence generation based on insertion operations. Unlike typical autoregressive models which rely on a fixed, often left-to-right ordering of the…

Let $\mathcal{G}$ be a Grothendieck category. We prove completeness of the Gorenstein injective cotorsion pair whenever $\mathcal{G}$ admits a set of Tate trivial generators, and show that having such generators is necessary for…

Recent efforts to unravel the mystery of implicit regularization in deep learning have led to a theoretical focus on matrix factorization -- matrix completion via linear neural network. As a step further towards practical deep learning, we…

We study ideal-simple commutative semirings and summarize the results giving their classification, in particular when they are finitely generated. In the principal case of (para)semifields, we then consider their minimal number of…

For any ring $R$, we investigate balanced pairs of classes of modules and their relations to cotorsion triples. We characterize the case when a balanced pair generates a tilting cotorsion pair, and dually, when it cogenerates a cotilting…

This paper presents an extension of the concept of NR-clean introduced in [12] to graded ring theory. We define and explore graded NR-clean rings, which generalize the class of graded U-nil clean previously studied in [15]. We provide…

This paper aims to develop a grafting method to address Majid's conjecture, which enables the construction of a larger target quantum group by grafting two given smaller ones. This method is significant for advancing the understanding of…

We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module…

We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called \textit{involutive semirings}. The semiring perspective helps us find a necessary and sufficient…

This note is a continuation of postgraduate thesis research carried out by the first author under the supervision of the second author at the University of Malta. In that research we took a look at several results relating Schur rings to…

In modern collider experiments, the quest to explore fundamental interactions between elementary particles has reached unparalleled levels of precision. Signatures from particle physics detectors are low-level objects (such as energy…