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.

舒尔的引理说，一个简单的左R模的内环是一个分割环。它对证明经典的环状结构定理（例如雅各布森密度定理和Wedderburn-Artin定理）起着基本作用。我们首先通过它们的\（\ Pi ^ {0} _ {1} \）子集定义简单左R-模的内同态环，并证明Schur的引理在\（\ mathrm RCA_ {0} \）中是可证明的。如果有忠实的简单左R-模块，则环R为左原始，如果有左规则模块\（_ {R} R \），则环R为左半简单是半简单的。Jacobson密度定理和Wedderburn-Artin定理分别描述了左原始环和左半简单环。然后，我们从逆数学的角度研究此类定理。