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 $$\Pi ^{0}_{1}$$ Π 1 0 subsets and show that Schur’s Lemma is provable in $$\mathrm RCA_{0}$$ R C A 0 . A ring R is left primitive if there is a faithful simple left R -module and left semisimple if the left regular module $$_{R}R$$ 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.