  In this paper we study the concept of radical factorization in the context of
abstract ideal theory in order to obtain a unified approach to the theory of
factorization into radical ideals and elements in the literature of commutative
rings, monoids and ideal systems. Using this approach we derive new
characterizations of classes of rings whose ideals are a product of radical
ideals, and we obtain also similar characterizations for classes of ideal
systems in monoids and star ideals in integral domains.
