The computability, definability, and proof theory of Artinian rings

Abstract

We show that, in the context of Reverse Mathematics, WKL0 (Weak König's Lemma) implies the statement ART0 which says that every Artinian ring is Noetherian, over RCA0 (Recursive Comprehension Axiom). To achieve this goal, we prove a general Computable Full Structure Theorem for computable Artinian rings similar to the classical version found in most Algebra texts.