On the Existence of Infinite Monomial Division Chains with Finitely Many Indeterminates

Let R be a ring, and let \(\overrightarrow{X}=\{X_0,X_1,\ldots ,X_N\}\) , \(N\in {\mathbb {N}}\) , finitely many indeterminate variables. We introduce a combinatorial principle in this context called \({\textsf{MDC}}\) that produces, for any given infinite sequence of monomials \(Z_0,Z_1,Z_2,\ldots ,Z_k,\ldots \in R[\overrightarrow{X}],\ k\in {\mathbb {N}},\) of strictly increasing degree, an infinite subsequence \(Z_{k_0},Z_{k_1},\cdots ,Z_{k_n},\cdots ,\ n\in {\mathbb {N}},\) such that for each n we have that \(Z_{k_n}\) divides \(Z_{k_{n+1}}\) . We show that, in the context of Reverse Mathematics and Subsystems of Second-Order Arithmetic, \({\textsf{MDC}}\) is an arithmetical principle equivalent to \({\mathsf {B\varSigma _2}}+{\mathsf {WO({\mathbb {N}}^{\mathbb {N}})}}\) , where \({\mathsf {B\varSigma _2}}\) is a bounding principle for \(\varSigma _2\) formulas equivalent to the Infinite Pigeonhole Principle (over \({\mathsf {RCA_0}}\) ), and \({\mathsf {WO({\mathbb {N}}^{\mathbb {N}})}}\) asserts that finite sequences of natural numbers \({\mathbb {N}}^{\mathbb {N}}\) are well-ordered via the length-lexicographic ordering.