Strictly flat cyclic Frechet modules and approximate identities

Let A be a locally m-convex Frechet algebra. We give a necessary and sufficient condition for a cyclic Frechet A-module X = A(+)/I to be strictly flat, generalizing thereby a criterion of Helemskii and Sheinberg [7]. 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 rn-convex Frechet 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 Frechet algebra has a locally bounded a.i. if and only if it has a bounded a.i. Some applications to amenable Frechet algebras are also given.