Let TR be a 1-tilting module with tilting torsion pair (Gen T, F) in Mod-R. The following
conditions are proved to be equivalent:(1) T is pure projective;(2) Gen T is a definable …

Let T be an infinitely generated tilting module of projective dimension at most one over an
arbitrary associative ring A, and let B be the endomorphism ring of T. We prove that if T is …

For four wide classes of topological rings R, we show that all flat left R-contramodules have
projective covers if and only if all flat left R-contramodules are projective if and only if all left …

D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin $ K $-algebras
that are the endomorphism rings of tilting objects in hereditary abelian categories whose …

Based on Beligiannis's theory in [Beligiannis, A.(2000). Relative homological algebra and
purity in triangulated categories. J. Algebra 227 (1): 268–361], we introduce and study E …

In this paper, we mainly focus on extension-closed subcategories of extriangulated
categories. Let X be an extension-closed subcategory. We show that if C is X-projective and …

Tilting modules appear in a variety of ways: their first definition was in the context of finite
dimensional algebras over a field. Later, they appeared naturally in applications of the …

Let $\Lambda $ be a finite-dimensional algebra. A wide subcategory of $\mathsf
{mod}\Lambda $ is called left finite if the smallest torsion class containing it is functorially …

Given small dg categories A and B and a BA-bimodule T, we give necessary and sufficient
conditions for the associated derived functors of Hom and the tensor product to be fully …