Ring structure theorems and arithmetic comprehension


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.

This is a preview of subscription content, log in to check access.


  1. 1.

    Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules, 2nd edn., Graduate Texts in Mathematics, 13. Springer, New York (1992)

  2. 2.

    Conidis, C.J.: Chain conditions in computable rings. Trans. Am. Math. Soc. 362, 6523–6550 (2010)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Downey, R.G., Lempp, S., Mileti, J.R.: Ideals in commutative rings. J. Algebra 314, 872–887 (2007)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Downey, R.G., Hirschfeldt, D.R., Kach, A.M., Lempp, S., Mileti, J.R., Montalbán, A.: Subspaces of computable vector spaces. J. Algebra 314, 888–894 (2007)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Farb, B., Dennis, R.K.: Noncommutative Algebra. Graduate Texts in Mathematics, vol. 144. Springer, New York (1993)

    Google Scholar 

  6. 6.

    Friedman, H.M., Simpson, S.G., Smith, R.L.: Countable algebra and set existence axioms. Ann. Pure Appl. Logic 25, 141–181 (1983)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Lam, T.Y.: A First Course in Noncommutative Rings, 2nd edn. Graduate Texts in Mathematics, 131. Springer, New York (2001)

  8. 8.

    Sato, T.: Reverse Mathematics and Countable Algebraic Systems. Ph.D. thesis, Tohoku University, Sendai, Japan (2016)

  9. 9.

    Simpson, S.G.: Subsystems of Second Order Arithmetic. Springer, Berlin (1999)

    Google Scholar 

  10. 10.

    Wu, H.: The complexity of radicals and socles of modules. Notre Dame J. Form. Logic 61, 141–153 (2020)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Yamazaki, T.: Reverse Mathematics and Commutative Ring Theory. Computability Theory and Foundations of Mathematics, Tokyo Institute Of Technology, February 18–20 (2013)

Download references

Author information



Corresponding author

Correspondence to Huishan Wu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation


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

Mathematics Subject Classification

  • 03B30
  • 03D15