@article{0193bb38-f0c1-73ec-be4b-ed4f389c6893,
author = {Huishan Wu},
title = {Effective aspects of semiperfect rings},
journal = {Reports on Mathematical Logic},
volume = {2024},
number = {Number 59},
year = {2024},
issn = {0137-2904},
pages = {3-26},keywords = {Reverse mathematics; semiperfect rings; perfect rings; local ring},
abstract = {This paper studies effective aspects of semiperfect rings from the standpoint of reverse mathematics. Based on first-order Jacobson radicals of rings, we define a ring R with the Jacobson radical Jac(R) to be semiperfect if the quotient ring R/Jac(R) is semisimple, and idempotents of the quotient ring can be lifted to R. Using elementary matrix operations in linear algebra, we show that RCA0 proves a characterization of semiperfect rings in terms of idempotents of rings. Semiperfect rings are generalizations of semisimple rings and local rings, and semiperfect rings R with R/Jac(R) simple are isomorphic to matrix rings over local rings. Based on the effective characterization of semiperfect rings via idempotents, we prove the structure theorem of semiperfect rings R with R/Jac(R) simple in RCA0. Left perfect rings or right perfect rings are always semiperfect. Finally, we provide a proof for the structure theorem of one-sided perfect rings R with R/Jac(R) simple in WKL0.},
doi = {10.4467/20842589RM.24.001.20696},
url = {https://ejournals.eu/en/journal/reports-on-mathematical-logic/article/effective-aspects-of-semiperfect-rings}
}