On the Category of Ideals of Ring

In this paper we describe the category whose objects are principal ideals of a ring and morphisms are appropriate translations and it is shown that such a category is an abelian category. Further we discuss various properties of such categories.

Semi-derived Ringel-Hall algebras and Hall algebras of odd-periodic relative derived categories

Let t be a positive integer and A be a hereditary abelian category satisfying some finiteness conditions.We define the semi-derived Ringel-Hall algebra of A from the category C_(Z/t)(A)of Z/t-graded complexes and obtain a natural basis of the semi-derived Ringel-Hall algebra.Moreover,we describe the semiderived Ringel-Hall algebra by the generators and defining relations.In particular,if t is an odd integer,we show an embedding of the derived Hall algebra of the odd-periodic relative derived category in the extended semi-derived Ringel-Hall algebra.

Phases of Stable Representations of Euclidean and Wild Quivers

We consider stable representations of non-Dynkin quivers with respect to a central charge.These attract a lot of interest in mathematics and physics since they can be identified with so-called BPS states.Another motivation is the work of Dimitrov et al.on the phases of stable representations of the generalized Kronecker quiver.One aim is to explain for general Euclidean and wild quivers the behavior of phases of stable representations well known in some examples.In addition,we study especially the behavior of preinjective,postprojective and regular indecomposable modules.We show that the existence of a stable representation with self-extensions implies the existence of infinitely many stables without self-extensions for rigid central charges.In this case the phases of the stable representations approach one or two limit points.In particular,the phases are not dense in two arcs.The category of representations of acyclic quivers is a special case of rigid Abelian categories which show this behavior for rigid central charges.

(D_(4))-Objects in Abelian Categories

Let A be an abelian category and M∈A.Then M is called a(D_(4))-object if,whenever A and B are subobjects of M with M=A⊕B and f:A→B is an epimorphism,Ker f is a direct summand of A.In this paper we give several equivalent conditions of(D_(4))-objects in an abelian category.Among other results,we prove that any object M in an abelian category A is(D_(4))if and only if for every subobject K of M such that K is the intersection K_(1)∩K_(2)of perspective direct summands K_(1)and K_(2)of M with M=K_(1)+K_(2),every morphismφ:M→M/K can be lifted to an endomorphismθ:M→M in End _(A)(M).

Torsion pairs in recollements of abelian categories

For a recollement （A,B,C） of abelian categories, we show that torsion pairs in A and C can induce torsion pairs in B; and the converse holds true under certain conditions.

The Category of Modules on an n-Trivial Extension: the Basic Properties

In this paper we investigate a categorical aspect of n-trivial extension of a ring by a family of modules.Namely,we introduce the right(resp.,left)n-trivial extension of a category by a family of endofunctors.Among other results,projective,injective and flat objects of this category are characterized,and two applications are presented at the end of this paper.We characterize when an n-trivial extension ring is k-perfect and establish a result on the self-injective dimension of an n-trivial extension ring.

Reduction of Wide Subcategories and Recollements

In this paper,we prove a reduction result on wide subcategories of abelian categories which is similar to the Calabi-Yau reduction,silting reduction andτ-tilting reduction.More precisely,if an abelian category A admits a recollement relative to abelian categories A'and A'',which is denoted by(A',A,A'',i^(*),i_(*),i^(!),j_(!),j^(*),j_(*)),then the assignment C→j^(*)(C)defines a bijection between wide subcategories in A containing i_(*)(A')and wide subcategories in A''.Moreover,a wide subcategory C of A containing i_(*)(A')admits a new recollement relative to A'and j_(*)(C)which is induced from the original recollement.

Special precovering classes in comma categories

Let T be a right exact functor from an abelian category B into another abelian category A.Then there exists a functor p from the product category A×B to the comma category(T↓A).In this paper,we study the property of the extension closure of some classes of objects in(T↓A),the exactness of the functor p and the detailed description of orthogonal classes of a given class p(X,Y)in(T↓A).Moreover,we characterize when special precovering classes in abelian categories A and B can induce special precovering classes in(T↓A).As an application,we prove that under suitable conditions,the class of Gorenstein projective leftΛ-modules over a triangular matrix ringΛ=(R M 0 S)is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering.Consequently,we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.