  In this paper we are concerned with the finiteness property of Ext-indices of
several ring extensions. In this direction, we introduce some conjectures and
discuss the relationship of them. Also we give affirmative answers to these
conjectures in some special cases. Furthermore, we prove that the trivial
extension of an Artinian local ring by its residue class field is always of
finite Ext-index and we show that the Auslander-Reiten conjecture is true for
this type of rings.
