Reverse mathematics of rings

TL;DR

This paper applies reverse mathematics to determine the axioms needed for key results in commutative ring theory, analyzing concepts like primary ideals, Noetherian conditions, and various integral domains.

Contribution

It introduces the first reverse mathematics analysis of primary ideals, radical of ideals, and different Noetherian definitions within weak base systems.

Findings

Primary ideals and radicals require specific axioms in reverse mathematics.

Different Noetherian definitions have distinct logical strengths.

Systematic study of PIDs, UFDs, Bézout, and GCD domains in reverse mathematics.

Abstract

Using the tools of reverse mathematics in second-order arithmetic, as developed by Friedman, Simpson, and others, we determine the axioms necessary to develop various topics in commutative ring theory. Our main contributions to the field are as follows. We look at fundamental results concerning primary ideals and the radical of an ideal, concepts previously unstudied in reverse mathematics. Then we turn to a fine-grained analysis of four different definitions of Noetherian in the weak base system $\mathsf{RCA}_0 + \mathsf{I}\Sigma_2$. Finally, we begin a systematic study of various types of integral domains: PIDs, UFDs and B\'ezout and GCD domains.