Tilting pairs in extriangulated categories

Extriangulated categories were introduced by Nakaoka and Palu to give a unification of properties in exact categories and extension-closed subcategories of triangulated categories. A notion of tilting pairs in an extriangulated category is introduced in this paper. We give a Bazzoni characterization of tilting pairs in this setting. We also obtain the Auslander–Reiten correspondence of tilting pairs which classifies finite $\mathcal {C}$ -tilting subcategories for a certain self-orthogonal subcategory $\mathcal {C}$ with some assumptions. This generalizes the known results given by Wei and Xi for the categories of finitely generated modules over Artin algebras, thereby providing new insights in exact and triangulated categories.

ON INTERMEDIATE RINGS WHICH ARE FINITELY GENERATED MODULES OVER A NOETHERIAN RING

Cho (R, m) là vành giao hoán Noether và Q(R) là vành các thương toàn phần của R. Mục đích của bài báo này là nghiên cứu cấu trúc của các vành trung gian giữa R và Q(R). Gọi X là tập tất cả các lớp tương đương [I], trong đó I là ideal của R sao cho I 2 = aI với a ∈ I là phần tử không là ước của không trong R. Gọi Y là tập tất cả các vành trung gian A giữa R và Q(R) sao cho A là R-môđun hữu hạn sinh. Trong bài báo này, chúng tôi thiết lập một song ánh từ X đến Y. Một số ví dụ được đưa ra để làm rõ kết quả. Thứ nhất, chúng tôi chỉ ra nếu R là một miền ideal chính thì R là phần tử duy nhất của Y. Thứ hai, cho một vành Buchsbaum R mà không là Cohen-Macaulay, chúng tôi xây dựng một vành trung gian Cohen-Macaulay A ∈ Y. Để giải quyết vấn đề, chúng tôi áp dụng phương pháp nghiên cứu của S. Goto năm 1983, L. T. Nhàn và M. Brodmann 2012.

Lattices over Bass Rings and Graph Agglomerations

We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory of the corresponding monoid T(R). Results of Levy–Wiegand and Levy–Odenthal together with a study of the local case yield an explicit description of T(R). The monoid is typically neither factorial nor cancellative. Nevertheless, we construct a transfer homomorphism to a monoid of graph agglomerations—a natural class of monoids serving as combinatorial models for the factorization theory of T(R). As a consequence, the monoid T(R) is transfer Krull of finite type and several finiteness results on arithmetical invariants apply. We also establish results on the elasticity of T(R) and characterize when T(R) is half-factorial. (Factoriality, that is, torsion-free Krull–Remak–Schmidt–Azumaya, is characterized by a theorem of Levy–Odenthal.) The monoids of graph agglomerations introduced here are also of independent interest.

Strongly D2 modules and their applications

An [Formula: see text]-module [Formula: see text] is called strongly [Formula: see text] if [Formula: see text] is a [Formula: see text] (equivalently, direct projective) module for every positive integer [Formula: see text]. In this paper, we consider the class of quasi-projective [Formula: see text]-modules, the class of strongly [Formula: see text] [Formula: see text]-modules and the class of [Formula: see text]-modules. We first show that these classes are distinct, which gives a negative answer to the question raised by Li–Chen–Kourki. We also give structural characterizations of strongly [Formula: see text] modules for finitely generated modules over a principal ideal domain. In addition, we characterize some rings such as Artinian semisimple rings, hereditary rings, semihereditary rings and perfect rings in terms of strongly [Formula: see text] modules.

𝐿-dimension for modules over a local ring

We introduce a new homological dimension for finitely generated modules over a commutative local ring R R , which is based on a complex derived from a free resolution L L of the residue field of R R , and called L L -dimension. We prove several properties of L L -dimension, give some applications, and compare L L -dimension to complete intersection dimension.

A characterization of right 4-Nakayama artin algebras

In this paper, we study the category of finitely generated modules over a class of right [Formula: see text]-Nakayama artin algebras. This class of algebras appear naturally in the study of representation-finite artin algebras. First, we give a characterization of right [Formula: see text]-Nakayama artin algebras. Then, we classify finitely generated indecomposable right modules over right [Formula: see text]-Nakayama artin algebras. We also compute almost split sequences for the class of right [Formula: see text]-Nakayama artin algebras.

Similarity Classes of Linear Transformations

In this paper, we investigate the similarity classes of linear transformations on a vector space using structure theorem for finitely generated modules over a principal ideal domain. We also establish formulae to count similarity classes with a given polynomial as a characteristic polynomial and to count total number of classes when the scalar field is finite.