Generalized injectivity
Generalized projectivity
Generalized projectivity. II.
Generation of preradicals
Gorenstein star modules and Gorenstein tilting modules
We introduce the notion of Gorenstein star modules and obtain some properties and a characterization of them. We mainly give the relationship between -Gorenstein star modules and -Gorenstein tilting modules, see L. Yan, W. Li, B. Ouyang (2016), and a new characterization of -Gorenstein tilting modules.
Hereditary and cohereditary preradicals
Hereditary tensor-orthogonal theories
Hereditary torsion classes of S-systems.
Idempotent completion of pretriangulated categories
A pretriangulated category is an additive category with left and right triangulations such that these two triangulations are compatible. In this paper, we first show that the idempotent completion of a left triangulated category admits a unique structure of left triangulated category and dually this is true for a right triangulated category. We then prove that the idempotent completion of a pretriangulated category has a natural structure of pretriangulated category. As an application, we show that...
Lattices of orthogonal theories
Localization and colocalization in tilting torsion theory for coalgebras
Tilting theory plays an important role in the representation theory of coalgebras. This paper seeks how to apply the theory of localization and colocalization to tilting torsion theory in the category of comodules. In order to better understand the process, we give the (co)localization for morphisms, (pre)covers and special precovers. For that reason, we investigate the (co)localization in tilting torsion theory for coalgebras.
Nilpotence, radicaux et structures monoïdales
Non-singular covers over monoid rings
We shall introduce the class of strongly cancellative multiplicative monoids which contains the class of all totally ordered cancellative monoids and it is contained in the class of all cancellative monoids. If is a strongly cancellative monoid such that for each and if is a ring such that for each , then the class of all non-singular left -modules is a cover class if and only if the class of all non-singular left -modules is a cover class. These two conditions are also equivalent whenever...
Non-singular covers over ordered monoid rings
Let be a multiplicative monoid. If is a non-singular ring such that the class of all non-singular -modules is a cover class, then the class of all non-singular -modules is a cover class. These two conditions are equivalent whenever is a well-ordered cancellative monoid such that for all elements with there is such that . For a totally ordered cancellative monoid the equalities and hold, being Goldie’s torsion theory.
Non-singular precovers over polynomial rings
One of the results in my previous paper On torsionfree classes which are not precover classes, preprint, Corollary 3, states that for every hereditary torsion theory for the category -mod with , being Goldie’s torsion theory, the class of all -torsionfree modules forms a (pre)cover class if and only if is of finite type. The purpose of this note is to show that all members of the countable set of rings have the property that the class of all non-singular left modules forms a (pre)cover...
On generation of torsion theories
On generation of torsion theories: a correction
On Kurosh-Amitsur radicals of finite groups.
On torsionfree classes which are not precover classes
In the class of all exact torsion theories the torsionfree classes are cover (precover) classes if and only if the classes of torsionfree relatively injective modules or relatively exact modules are cover (precover) classes, and this happens exactly if and only if the torsion theory is of finite type. Using the transfinite induction in the second half of the paper a new construction of a torsionfree relatively injective cover of an arbitrary module with respect to Goldie’s torsion theory of finite...
Präradikale und Endomorphismenringe von Moduln.