Related papers: Strongly CS-Rickart and dual strongly CS-Rickart o…
We introduce (dual) relative CS-Rickart objects in abelian categories, as common generalizations of (dual) relative Rickart objects and extending (lifting) objects. We study direct summands and (co)products of (dual) relative CS-Rickart…
We introduce and study (dual) strongly relative Rickart objects in abelian categories. We prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to…
We show how the theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories. For each of them, we prove general…
We investigate relative CS-Baer objects in abelian categories in relationship with other relevant classes of objects such as relative Baer objects, extending objects, objects having certain summand intersection properties and relative…
We introduce and investigate (dual) relative split objects with respect to a fully invariant short exact sequence in abelian categories. We compare them with (dual) relative Rickart objects, and we study their behaviour with respect to…
We study the transfer of (dual) relative CS-Rickart properties via functors between abelian categories. We consider fully faithful functors as well as adjoint pairs of functors. We give several applications to Grothendieck categories and,…
We study the transfer of (dual) relative Rickart properties via functors between abelian categories, and we deduce the transfer of (dual) relative Baer property. We also give applications to Grothendieck categories, comodule categories and…
Recently Dupont proved that the categories of discrete and codiscrete (or connected) objects in an abelian 2-category are equivalent abelian categories. He posses also a question whether any abelian category comes in this way. We will give…
We relativize the notion of a compact object in an abelian category with respect to a fixed subclass of objects. We show that the standard closure properties persist to hold in this case. Furthermore, we describe categorical and…
We establish connections between silting and tilting objects in an abelian category $\mathcal{B}$ and those in a cleft extension $\mathcal{A}$ of $\mathcal{B}$, which provides a method for constructing more silting and tilting objects. Then…
Let $\mathcal{A}$ be an abelian category. For a pair $(\mathcal{X},\mathcal{Y}$ of classes of objects in $\mathcal{A},$ we define the weak and the $(\mathcal{X},\mathcal{Y})$-Gorenstein relative projective objects in $\mathcal{A}$. We point…
We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair $(\mathcal{A},\mathcal{B})$ in an exact category $\mathcal{C}$, $\mathcal{A}$ coincides…
In this paper we develop a duality theory for all finite-dimensional near-vector spaces and introduce a notion of inner product tailored to the broad and natural class of strongly regular near-vector spaces. This generalized construction…
We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely…
We introduce the notions of proto-complete, complete, complete* and strong-complete objects in pointed categories. We show under mild conditions on a pointed exact protomodular category that every proto-complete (respectively complete)…
In this paper, we dualize the concept of {\Sigma}-Rickart modules as {\Sigma}-dual Rickart modules. An R-module M is said to be {\Sigma}-dual Rickart if the direct sum of arbitrary copies of M is dual Rickart. We prove that each…
We study the properties of the relative derived category $D_{\mathscr{C}}^{b}$($\mathscr{A}$) of an abelian category $\mathscr{A}$ relative to a full and additive subcategory $\mathscr{C}$. In particular, when $\mathscr{A}=A{\text -}\mod$…
For a finite abelian group action on a linear category, we study the dual action given by the character group acting on the category of equivariant objects. We prove that the groups of equivariant autoequivalences on these two categories…
In an ongoing project to classify all hereditary abelian categories, we provide a classification of Ext-finite directed hereditary abelian categories satisfying Serre duality up to derived equivalence. In order to prove the classification,…
We extend to semi-abelian categories the notion of characteristic subobject, which is widely used in group theory and in the theory of Lie algebras. Moreover, we show that many of the classical properties of characteristic subgroups of a…