When is every idempotent preradical of R-Mod a hereditary radical?

In this work, we study idempotent preradicals with the properties of being hereditary, radical, and stable. We prove that if every idempotent preradical is radical, then every semiartinian left R -module is semisimple. Also, if every idempotent preradical is hereditary, then R is a left self-injective and a left MAX MAX ring. And, if every idempotent preradical is stable, then R is a left V -ring and a left Noetherian ring. Finally, we characterize the rings where every idempotent preradical is a hereditary radical; in the commutative case R must be semisimple, and in the noncommutative case R must be semisimple and must also satisfy that the left annihilator of every element in R is a two sided ideal of R .