  Let A be a locally m-convex Fr\'echet algebra. We give a necessary and
sufficient condition for a cyclic Fr\'echet A-module X=A_+/I to be strictly
flat, generalizing thereby a criterion of Helemskii and Sheinberg. To this end,
we introduce a notion of locally bounded approximate identity (a.i.), and we
show that X is strictly flat if and only if the ideal I has a right locally
bounded a.i. An example is given of a commutative locally m-convex Fr\'echet
algebra that has a locally bounded a.i., but does not have a bounded a.i. On
the other hand, we show that a quasinormable locally m-convex Fr\'echet algebra
has a locally bounded a.i. if and only if it has a bounded a.i. Some
applications to amenable Fr\'echet algebras are also given.
