  In this paper we explore consequences of the vanishing of ${\rm Ext}$ 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.
