Abstract
This paper studies the structure of semisimple rings using techniques of reverse mathematics, where a ring is left semisimple if the left regular module is a finite direct sum of simple submodules. The structure theorem of left semisimple rings, also called Wedderburn-Artin Theorem, is a famous theorem in noncommutative algebra, says that a ring is left semisimple if and only if it is isomorphic to a finite direct product of matrix rings over division rings. We provide a proof for the theorem in \(\mathrm RCA_{0}\), showing the structure theorem for computable semisimple rings. The decomposition of semisimple rings as finite direct products of matrix rings over division rings is unique. Based on an effective proof of the Jordan-Hölder Theorem for modules with composition series, we also provide an effective proof for the uniqueness of the matrix decomposition of semisimple rings in \(\mathrm RCA_{0}\).
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The author sincerely appreciates the referees for invaluable comments and useful suggestions. This work is supported by the Science Foundation of Beijing Language and Culture University (supported by “the Fundamental Research Funds for the Central Universities”) (Grant No. 23YJ080006).
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Wu, H. Structure of semisimple rings in reverse and computable mathematics. Arch. Math. Logic 62, 1083–1100 (2023). https://doi.org/10.1007/s00153-023-00885-3
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DOI: https://doi.org/10.1007/s00153-023-00885-3