Ring Constructions and Generation of the Unbounded Derived Module Category

Abstract We consider the smallest triangulated subcategory of unbounded derived module category a ring that contains injective modules and is closed under set indexed coproducts. If this entire category, then we say injectives generate for ring. In particular, ask whether, if collection rings, do related constructions, vice versa. provide sufficient conditions statement to hold various constructions including recollements, extensions equivalences.