  In this paper we describe three different variations of prime ideals:
strongly irreducible ideals, strongly prime ideals and insulated prime ideals
in the context of Leavitt path algebras. We give necessary and sufficient
conditions under which a proper ideal of a Leavitt path algebra $L$ is a
product as well as an intersection of finitely many of these different types of
prime ideals. Such factorizations, when they are irredundant, are shown to be
unique except for the order of the factors. We also characterize the Leavitt
path algebras $L$ in which every ideal admits such factorizations and also in
which every ideal is one of these special type of ideals.
