Reverse mathematics and semisimple rings

This paper studies various equivalent characterizations of left semisimple rings from the standpoint of reverse mathematics. We first show that A C A 0 is equivalent to the statement that any left module over a left semisimple ring is semisimple over R C A 0 . We then study characterizations of left...

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Bibliographic Details
Published inArchive for mathematical logic Vol. 61; no. 5-6; pp. 769 - 793
Main Author Wu, Huishan
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2022
Springer Nature B.V
Subjects
Algebra
Equivalence
Mathematical analysis
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Modules
Rings (mathematics)
03B30
03D15
Reverse mathematics
Projective modules
Semisimple rings
Injective modules
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Summary:This paper studies various equivalent characterizations of left semisimple rings from the standpoint of reverse mathematics. We first show that A C A 0 is equivalent to the statement that any left module over a left semisimple ring is semisimple over R C A 0 . We then study characterizations of left semisimple rings in terms of projective modules as well as injective modules, and obtain the following results: (1) A C A 0 is equivalent to the statement that any left module over a left semisimple ring is projective over R C A 0 ; (2) A C A 0 is equivalent to the statement that any left module over a left semisimple ring is injective over R C A 0 ; (3) R C A 0 proves the statement that if every cyclic left R -module is projective, then R is a left semisimple ring; (4) A C A 0 proves the statement that if every cyclic left R -module is injective, then R is a left semisimple ring.
ISSN:0933-5846
1432-0665
DOI:10.1007/s00153-021-00812-4