# Ring Constructions and Generation of the Unbounded Derived Module Category

## TL;DR

This paper investigates conditions under which injective modules generate the entire unbounded derived category of a ring, exploring how this property behaves under various ring constructions and extensions.

## Contribution

It provides sufficient conditions for injectives to generate derived categories across different ring constructions, advancing understanding of their generative properties.

## Key findings

- Injectives generate the derived category under certain conditions.
- Generation property is preserved under specific ring extensions.
- Results apply to recollements and module category equivalences.

## Abstract

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