Classification of the completely primary totally ramified orders of finite lattice type.
We continue the study started recently by Agore, Bontea and Militaru in “Classifying bicrossed products of Hopf algebras” (2014), by describing and classifying all Hopf algebras that factorize through two Sweedler’s Hopf algebras. Equivalently, we classify all bicrossed products . There are three steps in our approach. First, we explicitly describe the set of all matched pairs by proving that, with the exception of the trivial pair, this set is parameterized by the ground field . Then, for...
A matrix is -clean provided there exists an idempotent such that and . We get a general criterion of -cleanness for the matrix . Under the -stable range condition, it is shown that is -clean iff . As an application, we prove that the -cleanness and unit-regularity for such matrix over a Dedekind domain coincide for all . The analogous for property is also obtained.
It is well known that a semigroup S is a Clifford semigroup if and only if S is a strong semilattice of groups. We have recently extended this important result from semigroups to semirings by showing that a semiring S is a Clifford semiring if and only if S is a strong distributive lattice of skew-rings. In this paper, we introduce the notions of Clifford semidomain and Clifford semifield. Some structure theorems for these semirings are obtained.
Si considerano le estensioni chiuse di un -modulo mediante un -modulo nel caso in cui sia un anello semi-artiniano, cioè un anello con la proprietà che per ogni quoziente sia soc . Tali estensioni sono caratterizzate dal fatto che deve essere un sottomodulo semi-puro di .
We consider rings equipped with a closure operation defined in terms of a collection of commuting idempotents, generalising the idea of a topological closure operation defined on a ring of sets. We establish the basic properties of such rings, consider examples and construction methods, and then concentrate on rings which have a closure operation defined in terms of their lattice of central idempotents.
Let be a field and a finite-dimensional -algebra of global dimension . We construct a triangulated category associated to which, if is hereditary, is triangle equivalent to the cluster category of . When is Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our results also...
Starting from an arbitrary cluster-tilting object in a 2-Calabi–Yau triangulated category over an algebraically closed field, as in the setting of Keller and Reiten, we define, for each object , a fraction using a formula proposed by Caldero and Keller. We show that the map taking to is a cluster character, i.e. that it satisfies a certain multiplication formula. We deduce that it induces a bijection, in the finite and the acyclic case, between the indecomposable rigid objects of the cluster...
We define a graded twisted-coassociative coproduct on the tensor algebra the desuspension space of a graded vector space . The coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1 with sequences of multilinear maps on (resp. graded Loday structures on , sequences that we call Loday infinity structures on ). We prove a minimal model theorem for Loday infinity algebras and observe that the category contains the category as...
We develop a technique for the study of K-coalgebras and their representation types by applying a quiver technique and topologically pseudocompact modules over pseudocompact K-algebras in the sense of Gabriel [17], [19]. A definition of tame comodule type and wild comodule type for K-coalgebras over an algebraically closed field K is introduced. Tame and wild coalgebras are studied by means of their finite-dimensional subcoalgebras. A weak version of the tame-wild dichotomy theorem of Drozd [13]...
Let (A,α) and (B,β) be two Hom-Hopf algebras. We construct a new class of Hom-Hopf algebras: R-smash products . Moreover, necessary and sufficient conditions for to be a cobraided Hom-Hopf algebra are given.