Reverse mathematics and semisimple rings

Abstract

This paper studies various equivalent characterizations of left semisimple rings from the standpoint of reverse mathematics. We first show that $\mathrm ACA_{0}$ is equivalent to the statement that any left module over a left semisimple ring is semisimple over $\mathrm RCA_{0}$. We then study characterizations of left semisimple rings in terms of projective modules as well as injective modules, and obtain the following results: (1) $\mathrm ACA_{0}$ is equivalent to the statement that any left module over a left semisimple ring is projective over $\mathrm RCA_{0}$; (2) $\mathrm ACA_{0}$ is equivalent to the statement that any left module over a left semisimple ring is injective over $\mathrm RCA_{0}$; (3) $\mathrm RCA_{0}$ proves the statement that if every cyclic left R-module is projective, then R is a left semisimple ring; (4) $\mathrm ACA_{0}$ proves the statement that if every cyclic left R-module is injective, then R is a left semisimple ring.