Ring structure theorems and arithmetic comprehension

Abstract

Schur’s Lemma says that the endomorphism ring of a simple left R-module is a division ring. It plays a fundamental role to prove classical ring structure theorems like the Jacobson Density Theorem and the Wedderburn–Artin Theorem. We first define the endomorphism ring of simple left R-modules by their \(\Pi ^{0}_{1}\) subsets and show that Schur’s Lemma is provable in \(\mathrm RCA_{0}\). A ring R is left primitive if there is a faithful simple left R-module and left semisimple if the left regular module \(_{R}R\) is semisimple. The Jacobson Density Theorem and the Wedderburn-Artin Theorem characterize left primitive ring and left semisimple ring, respectively. We then study such theorems from the standpoint of reverse mathematics.

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Correspondence to Huishan Wu.

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This work is supported by the National Natural Science Foundation of China (No. 61972052), the Discipline Team Support Program of Beijing Language and Culture University (No. GF201905), and the Science Foundation of Beijing Language and Culture University (supported by “the Fundamental Research Funds for the Central Universities”) (Grant No. 19YJ040009).

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Wu, H. Ring structure theorems and arithmetic comprehension . Arch. Math. Logic (2020). https://doi.org/10.1007/s00153-020-00738-3

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Keywords

  • Reverse mathematics
  • Schur’s Lemma
  • The Jacobson Density Theorem
  • The Wedderburn–Artin Theorem

Mathematics Subject Classification

  • 03B30
  • 03D15