Monomorphisms in the category of firm modules

Abstract

In this article, we consider the category of unitary right modules over an (associative) ring and the category of firm right modules over an idempotent ring. We study monomorphisms in these categories and give conditions under which morphisms are monomorphisms in the category of firm modules. We also prove that the lattice of categorically defined subobjects of a firm module is isomorphic to the lattice of unitary submodules of that module.     

Stable model categories are categories of modules

A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R.     

Coherent rings of finite weak global dimension

The category of left modules over right coherent rings of finite weak global dimension has several nice features. For example, every left module over such a ring has a flat cover (Belshoff, Enochs, Xu) and, if the weak global dimension is at most two, every left module has a flat envelope (Asensio, Martínez). We will exploit these features of this category to study its objects. In particular, we will consider orthogonal complements (relative to the extension functor) of several classes of modules in this category. In the case of a commutative ring we describe an idempotent radical on its category of modules which, when the weak global dimension does not exceed 2, can be used to analyze the structure of the flat envelopes and of the ring itself.

     

Separable Groupoid Rings

We show that groupoid rings are separable over their ring of coefficients if and only if the groupoid is finite and the orders of the associated principal groups are invertible in the ring of coefficients. We use this to show that if we are given a finite groupoid, then the associated groupoid ring is semisimple (or hereditary) if and only if the ring of coefficients is semisimple (or hereditary) and the orders of the principal groups are invertible in the ring of coefficients. To this end, we extend parts of the theory of graded rings and modules from the group graded case to the category graded, and, hence, groupoid graded situation. In particular, we show that strongly groupoid graded rings are separable over their principal components if and only if the image of the trace map contains the identity.     

Characterization of eventually periodic modules in the singularity categories

The singularity category of a ring makes only the modules of finite projective dimension vanish among the modules, so that the singularity category is expected to characterize a homological property of modules of infinite projective dimension. In this paper, among such modules, we deal with eventually periodic modules over a left artin ring, and, as our main result, we characterize them in terms of morphisms in the singularity category. As applications, we first prove that, for the class of finite dimensional algebras over a field, being eventually periodic is preserved under singular equivalence of Morita type with level. Moreover, we determine which finite dimensional connected Nakayama algebras are eventually periodic when the ground field is algebraically closed.     

On the artinianness of the local homology modules

In [9] a local homology theory for Artinian modules over commutative rings, which is dual to the local cohomology theory for Noetherian modules, introduced and in [1] the main result of [9] extended. In this note we prove that the local homology modules of an Artinian module over a commutative ring (with identity) are Artinian.     

McCoy模的一些性质

设$R$是一个有单位元的环, $\mathcal{C}(R)$是右$R$模范畴. 在本文中, 我们介绍了semi-McCoy模的概念, 由此得到 $\mathcal{C}(R)$ 在满同态的核下封闭, 在一定条件下关于短正合列扩张以及直和也是封闭的. 我们同时也给出$\mathcal{C}(R[x])$和 $\mathcal{C}(R[x;x^{-1}])$子范畴的一些性质.     

$\mathcal{F}$-Perfect Rings and Modules

Let $R$ be a ring, and let $(\mathcal{F}, C)$ be a cotorsion theory. In this article, the notion of $\mathcal{F}$-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring $R$ is said to be right $\mathcal{F}$-perfect if $F$ is projective relative to $R$ for any $F ∈ \mathcal{F}$. We give some characterizations of $\mathcal{F}$-perfect rings. For example, we show that a ring $R$ is right $\mathcal{F}$-perfect if and only if $\mathcal{F}$-covers of finitely generated modules are projective. Moreover, we define $\mathcal{F}$-perfect modules and investigate some properties of them.     

On the cotorsion hull in torsion theory

The concept of cotorsion was first introduced in the category of Abelian groups (Fuchs [l] ). Matlis [5], studied the cotorsion modules over integral domains. Henderson and Orzech [4], Fuchs [2], and Mines [6], replaced the classical notion of torsion by a torsion theory (T,F) on R-mod, where R is not necessarily commutative ring. In this paper we find conditions on the torsion theory in order to get a T-cotorsion hull for every module. This generalizes the result of Fuchs [2].     

Irreducible maps of commutative rings

We give necessary conditions for a map to be irreducible (in the category of finitely generated, torsion free modules) over a non-local, commutative ring and sufficient conditions when the ring is Bass. In particular, we show that an irreducible map of ZG, where G is a square free abelian group, must be a monomorphism with a simple cokernel. We also show that local endomorphism rings are necessary and sufficient for the existence of almost split sequences over a commutative Bass ring and we explicitly describe the modules and the maps in those sequences. The results in this paper enable us to describe the Auslander-Reiten quiver of a non-local Bass ring in [8].     

Differential projective modules over differential rings

Differential modules over a commutative differential ring which are projective as ring modules, with differential homomorphisms, form an additive category. Every projective ring module is shown occurs as the underlying module of a differential module. Differential modules, projective as ring modules, are shown to be direct summands of differential modules free as ring modules; those which are differential direct summands of differential direct sums of the ring being induced from the subring of constants. Every differential module finitely generated and projective as a ring module is shown to have this form after a faithfully flat finitely presented differential extension of the base.     

Classifying thick subcategories of the stable category of Cohen-Macaulay modules

Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher-dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p-group due to Benson, Carlson and Rickard, we consider classifying thick subcategories of the stable category of Cohen-Macaulay modules over a Gorenstein local ring. The main result of this paper yields a complete classification of the thick subcategories of the stable category of Cohen-Macaulay modules over a local hypersurface in terms of specialization-closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus.     

ON ALMOST PRIME SUBMODULES

In this article,we define almost prime submodules as a new generalization of prime and weakly prime submodules of unitary modules over a commutative ring with identity.We study some basic properties of...     

Generators and Dimensions of Derived Categories of Modules

Several years ago, Bondal, Rouquier, and Van den Bergh introduced the notion of the dimension of a triangulated category, and Rouquier proved that the bounded derived category of coherent sheaves on a separated scheme of finite type over a perfect field has finite dimension. In this article, we study the dimension of the bounded derived category of finitely generated modules over a commutative Noetherian ring. The main result of this article asserts that it is finite over a complete local ring containing a field with perfect residue field. Our methods also give a ring-theoretic proof of the affine case of Rouquier's theorem.     

Locally Stable Grothendieck Categories: Applications

We introduce for any Grothendieck category the notion of stable localizing subcategory, as a localizing subcategory that can be written as an intersection of localizing subcategories defined by indecomposable injectives. A Grothendieck category in which every localizing subcategory is stable is called a locally stable category. As a main result we give a characterization of these categories in terms of the local stability of a localizing subcategory and its quotient category. The locally coirreducible categories (in particular, the categories with Gabriel dimension) and the locally noetherian categories are examples of locally stable categories. We also present some applications to the category of modules over a left fully bounded noetherian ring, to the category of comodules over a coalgebra and to the category of modules over graded rings.     

有限生成亚投射模范畴

对一个QF环R,本文证明:其投射左R模范畴是因式分解范畴当且仅当gl.dim R≤1．进一步,若 P(_RR)＝P(R_R)＝0,则其通过左模而得到的亚 Crothendieck群与其通过右模而得到的亚Grothendieck群在同构意义下是一样的．还证明了有限生成亚投射左R－模范畴不仅是一个因式分解范畴而且是一个带积的具有小的骨架子范畴的范畴．     

Triangulated Categories and the Ziegler Spectrum

The relationship between the Ziegler spectrum of (the category of modules over) a ring and the Ziegler spectrum of its derived category is investigated. Over von Neumann regular rings and hereditary rings the spectrum of the derived category is a disjoint union of copies of the spectrum of the ring but in general there are further indecomposable pure-injective objects of the derived category. Presented by A. Verschoren Mathematics Subject Classifications (2000) Primary: 18E30; secondary: 03C60.     

Separability and triangulated categories

We prove that the category of modules over a separable ring object in a tensor triangulated category admits a unique structure of triangulated category which is compatible with the original one. This applies in particular to étale algebras. More generally, we do this for exact separable monads.     

Solution to the Clebsch-Gordan problem for string algebras

The category of modules over a string algebra is equipped with a tensor product defined point-wise and arrow-wise in terms of the underlying quiver. In the present article we investigate how this tensor product interacts with the classification of indecomposables. We apply the results obtained to solve the Clebsch-Gordan problem for string algebras. Moreover, we describe the corresponding representation ring and tensor ideals in the module category.