The Auslander-Type Condition of Triangular Matrix Rings

TL;DR

This paper characterizes when a Noetherian ring satisfies a specific homological condition called the Auslander-type condition $G_n(k)$, showing it is equivalent for the ring and its associated lower triangular matrix rings.

Contribution

It establishes an equivalence of the Auslander-type condition $G_n(k)$ between a Noetherian ring and its lower triangular matrix rings of any degree.

Findings

Ring $R$ satisfies $G_n(k)$ iff its lower triangular matrix rings satisfy $G_n(k)$

Provides a characterization of Auslander-type conditions in matrix ring extensions

Extends understanding of homological properties in ring theory

Abstract

Let $R$ be a left and right Noetherian ring and $n,k$ any non-negative integers. $R$ is said to satisfy the Auslander-type condition $G_n(k)$ if the right flat dimension of the $(i+1)$-st term in a minimal injective resolution of $R_R$ is at most $i+k$ for any $0 \leq i \leq n-1$. In this paper, we prove that $R$ is $G_n(k)$ if and only if so is a lower triangular matrix ring of any degree $t$ over $R$.