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.