Effective aspects of semiperfect rings

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 ri...

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Bibliographic Details
Published inReports on mathematical logic Vol. 59; no. 59; pp. 3 - 26
Main Author Wu, Huishan
Format Journal Article
LanguageEnglish
Published Kraków Wydawnictwo Uniwersytetu Jagiellońskiego 01.01.2024
Jagiellonian University Press
Jagiellonian University-Jagiellonian University Press
Subjects
Linear algebra
Logic
Quotients
Rings (mathematics)
Structural analysis
Theorems
local ring
Reverse mathematics
perfect rings
semiperfect rings
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Summary: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.
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ISSN:0137-2904
2084-2589
DOI:10.4467/20842589RM.24.001.20696