The weak density of the non-invertible elements of a commutative algebra

Let X be a commutative locally convex Hausdorff topological algebra with identity over a non-trivially valued field F. Let Mc denote the continuous nontrivial homomorphisms of X into F and M the set of all maximal ideals of X. If the spectrum of each element x in X is the set of scalars {f(x) | f ∈ Mc}, it is shown that the singular elements of X are weakly dense in X if and only if M is an infinite set.