Related papers: Corings with decomposition and semiperfect corings

We show the close connection between appearingly different Galois theories for comodules introduced recently in [J. G\'omez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, arXiv:math.RA/0509106.] and…

Let $(R,\frak m)$ be a commutative noetherian local ring. In this paper, we prove that if $\frak m$ is decomposable, then for any finitely generated $R$-module $M$ of infinite projective dimension $\frak m$ is a direct summand of (a direct…

We describe how some aspects of abstract localization on module categories have applications to the study of injective comodules over some special types of corings. We specialize the general results to the case of Doi-Koppinen modules,…

A left and right noetherian semiperfect ring R is known to be indecomposable if and only if its factor by the second power of Jacobson radical is. This characterisation is used to study simple R-modules in terms of their Ext groups. It is…

It is shown that any finite complete covering of a non-commutative algebra in the sense of Calow and Matthes (J. Geom. Phys. 32 (2000), 114--165) gives rise to a Galois coring.

Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study of theory of rings and modules. They provide useful technical tools for studying their structure. In this paper we define one of the dimensions called couniserial…

In this paper we introduce and investigate the notion of semiseparable functor. One of its first features is that it allows a novel description of separable and naturally full functors in terms of faithful and full functors, respectively.…

Purpose: To develop the algebraic foundation of finite commutative ternary $\Gamma$-semirings by identifying their intrinsic invariants, lattice organization, and radical behavior that generalize classical semiring and $\Gamma$-ring…

A module over a ring $R$ is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings.…

Let $R$ be a ring and let $\mathcal C$ be a small class of right $R$-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let $\mathcal V (\mathcal C)$ denote a set of representatives of isomorphism classes…

It is proved that, for a left hereditary ring, an arbitrary left module has a representation in the form of the direct sum of a stable left module and indecomposable projective left modules (if and only if an arbitrary left module has a…

In this paper, we characterize several properties of commutative notherian local rings in terms of the left perpendicular category of the category of finitely generated modules of finite projective dimension. As an application we prove that…

Starting from a comonad G on a category A, and a functor L : B -> A with a right adjoint R : A -> B, we will give a parametrization of the functors K from B to the category of all G-coalgebras that factorize throughout L in terms of…

A module $M$ is {called} stable if it has no nonzero projective direct summand. For a ring $ R $, we study conditions under which $R$-modules from certain classes decompose as a direct sum of a projective submodule and a stable submodule.…

We develop a functorial framework for the ideal theory of commutative semirings using coherent frames and spectral spaces. Two central constructions-the radical ideal functor and the $k$-radical ideal functor-are shown to yield coherent…

Let $R$ be a (possibly noncommutative) ring and let $\mathcal C$ be a class of finitely generated (right) $R$-modules which is closed under finite direct sums, direct summands, and isomorphisms. Then the set $\mathcal V (\mathcal C)$ of…

Let $M$ be a left module over a ring $R$ and $I$ an ideal of $R$. We call $(P, f)$ a (locally)projective $I$-cover of $M$ if $f$ is an epimorphism from $P$ to $M$, $P$ is (locally)projective, $Kerf\subseteq IP$, and whenever $P=Kerf+X$,…

Let $\mathcal{P}$ be the class of rings for which every indecomposable right module is pure-projective or pure-injective. When $R$ is a Noetherian local commutative ring of maximal ideal $P$, it is proven that $R\in\mathcal{P}$ if and only…

This book is an introductory course to basic commutative algebra with a particular emphasis on finitely generated projective modules. We adopt the constructive point of view, with which all existence theorems have an explicit algorithmic…

We study non-counital coalgebras and their dual non-unital algebras, and introduce the finite dual of a non-unital algebra. We show that a theory that parallels in good part the duality in the unital case can be constructed. Using this, we…