Finitistic Extension Degree

TL;DR

This paper introduces the finitistic extension degree of a ring, explores its finiteness conditions, and applies these findings to the Auslander-Reiten Conjecture and Gorenstein rings, advancing understanding in homological algebra.

Contribution

It defines the finitistic extension degree, proves the Auslander-Reiten Conjecture for rings with finite degree, and analyzes its behavior under common ring modifications.

Findings

Rings with finite finitistic extension degree satisfy the Auslander-Reiten Conjecture.

Such rings also have finite finitistic dimension.

The paper details how the finitistic extension degree changes under various ring transformations.

Abstract

We introduce the finitistic extension degree of a ring and investigate rings for which it is finite. The Auslander-Reiten Conjecture is proved for rings of finite finitistic extension degree and these rings are also shown to have finite finitistic dimension. We apply these results to better understand a gen- eralized version of the Auslander-Reiten Condition for Gorenstein rings. We also record how the finitistic extension degree behaves with respect to many change of ring procedures that arise frequently in the commutative setting.