Abstract

This paper studies effective aspects of Jacobson radicals of rings and their applications from the viewpoint of reverse mathematics. First, we propose four radicals of rings, showing that the first order (resp., second order) left and right Jacobson radical coincide in (resp., ). Second, we study Jacobson radicals in left (resp., right) local rings and show that the second order left and right Jacobson radical of left (resp., right) local rings coincide within. Third, we apply our results about Jacobson radicals of rings and local rings to study properties of left (resp., right) strongly indecomposable modules; furthermore, we study effective aspects of typical lemmas and theorems related to strongly indecomposable modules, and show that proves the Fitting Lemma as well as the Krull‐Schmidt‐Azumaya Theorem and that proves the Krull‐Schmidt Theorem.