Maximal ideals in Loc(M,F) (Q1118863)

The author shows the following: Let \({\mathcal F}\) be a codimension one transversely oriented foliation of a compact manifold M. Then any maximal ideal of the algebra of all smooth functions constant along the leaves of \({\mathcal F}\) is of the form \(\{\) f: \(f| L=0\}\), where L is a compact leaf of \({\mathcal F}\) such that there are no open saturated neighbourhoods V of L for which the union of all compact leaves of \({\mathcal F}\) is nowhere dense in V.