Auslander-Reiten and Huneke-Wiegand conjectures over quasi-fiber product rings

Abstract

In this paper we explore consequences of the vanishing of $\mathrm{Ext}$ for finitely generated modules over a quasi-fiber product ring $R$; that is, $R$ is a local ring such that $R/(\underline{x})$ is a non-trivial fiber product ring, for some regular sequence $\underline{x}$ of $R$. Equivalently, the maximal ideal of $R/(\underline{x})$ decomposes as a direct sum of two nonzero ideals. Gorenstein quasi-fiber product rings are AB-rings and are Ext-bounded. We show in Theorem 3.31 that quasi-fiber product rings satisfy a sharpened form of the Auslander-Reiten Conjecture. We also make some observations related to the Huneke-Wiegand conjecture for quasi-fiber product rings.