# Gorenstein $\pi[T]$-projectivity with respect to a tilting module

## TL;DR

This paper introduces Gorenstein $ ext{pi}[T]$-projective modules relative to a tilting module, explores their properties, and characterizes rings where all modules are Gorenstein $ ext{pi}[T]$-projective, especially on $T$-cocoherent rings.

## Contribution

It defines Gorenstein $ ext{pi}[T]$-projectivity relative to a tilting module and provides characterizations of rings where all modules exhibit this property.

## Key findings

- Gorenstein $ ext{pi}[T]$-projective modules are introduced and studied.
- Characterizations of rings with all modules Gorenstein $ ext{pi}[T]$-projective are provided.
- On $T$-cocoherent rings, Gorenstein $ ext{pi}[T]$-projectivity of all modules is equivalent to $ ext{pi}[T]$-projectivity of $ ext{sigma}[T]$-injective modules.

## Abstract

Let $T$ be a tilting module. In this paper, Gorenstein $\pi[T]$-projective modules are introduced and some of their basic properties are studied. Moreover, some characterizations of rings over which all modules are Gorenstein $\pi[T]$-projective are given. For instance, on the $T$-cocoherent rings, it is proved that the Gorenstein $\pi[T]$-projectivity of all $R$-modules is equivalent to the $\pi[T]$-projectivity of $\sigma[T]$-injective as a module.