Characterizing von Neumann Regular Rings in Reverse Mathematics

Abstract

A von Neumann regular ring, named after John von Neumann for his work related to continuous geometry and operator algebras, has various equivalent characterizations in terms of popular properties of rings and modules. This paper mainly focuses on effective aspects of characterizations of von Neumann regular rings, and studies such characterizations by techniques of reverse mathematics. For not necessarily commutative rings, we obtain that RCA0 proves that a ring R is von Neumann regular iff every Σ10 finitely generated left ideal of R is generated by an idempotent iff every Σ10 cyclic left R-module is Lam-divisible. For commutative rings, we obtain that over RCA0, WKL0 (resp., ACA0) is equivalent to the statement that a commutative ring R is von Neumann regular iff the localization of R at any prime ideal (resp., maximal ideal) exists and it is a field. Lastly, we show that ACA0 proves that a commutative ring R is von Neumann regular iff every simple R-module is Lam-divisible.