This paper studies the structure of semisimple rings using techniques of reverse mathematics, where a ring is left semisimple if the left regular module is a finite direct sum of simple submodules. The structure theorem of left semisimple rings, also called Wedderburn-Artin Theorem, is a famous theorem in noncommutative algebra, says that a ring is left semisimple if and only if it is isomorphic to a finite direct product of matrix rings over division rings. We provide a proof for the theorem in \(\mathrm RCA_{0}\), showing the structure theorem for computable semisimple rings. The decomposition of semisimple rings as finite direct products of matrix rings over division rings is unique. Based on an effective proof of the Jordan-Hölder Theorem for modules with composition series, we also provide an effective proof for the uniqueness of the matrix decomposition of semisimple rings in \(\mathrm RCA_{0}\).

本文使用逆向数学技术研究半单环的结构，如果左正则模是简单子模的有限直和，则环是左半单环。左半单环结构定理，也称为Wedderburn-Artin定理，是非交换代数中的一个著名定理，说一个环是左半单环当且仅当它同构于矩阵环在除环上的有限直积。我们在\(\mathrm RCA_{0}\)中提供定理的证明，展示了可计算半单环的结构定理。半单环分解为矩阵环在分割环上的有限直积是唯一的。基于组合级数模的Jordan-Hölder定理的有效证明，我们还提供了对\(\mathrm RCA_{0}\)中半单环矩阵分解唯一性的有效证明。