%0 Journal Article
%T Effective aspects of semiperfect rings
%A Wu, Huishan
%J Reports on Mathematical Logic
%V 2024
%R 10.4467/20842589RM.24.001.20696
%N Number 59
%P 3-26
%K Reverse mathematics, semiperfect rings, perfect rings, local ring
%@ 0137-2904
%D 2024
%U https://ejournals.eu/en/journal/reports-on-mathematical-logic/article/effective-aspects-of-semiperfect-rings
%X 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.