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- Title
Effective aspects of Jacobson radicals of rings.
- 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 RCA0 (resp., ACA0). 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 RCA0. 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 RCA0 proves the Fitting Lemma as well as the Krull‐Schmidt‐Azumaya Theorem and that ACA0 proves the Krull‐Schmidt Theorem.
- Subjects
JACOBSON radical; LOCAL rings (Algebra); REVERSE mathematics; RIGHT &; left (Political science); INDECOMPOSABLE modules
- Publication
Mathematical Logic Quarterly, 2021, Vol 67, Issue 4, p489
- ISSN
0942-5616
- Publication type
Article
- DOI
10.1002/malq.202100002