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 Bezout and GCD domains.
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