In this paper we study the structure of closed weakly dense ideals in Privalov spaces

(1 < p <

) of holomorphic functions on the disk

: |z| < 1. The space

with the topology given by Stoll's metric [21] becomes an F-algebra. N. Mochizuki [16] proved that a closed ideal in

is a principal ideal generated by an inner function. Consequently, a closed subspace E of

is invariant under multiplication by z if and only if it has the form

for some inner function I. We prove that if

is a closed ideal in

that is dense in the weak topology of

, then

is generated by a singular inner function. On the other hand, if

is a singular inner function whose associated singular measure

has the modulus of continuity

, then we prove that the ideal

is weakly dense in

. Consequently, for such singular inner function

, the quotient space

is an F-space with trivial dual, and hence

does not have the separation property.