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.