Properties of factorizations with successive lengths in one-dimensional local domains (Q847953)

Let \(D\) be a \(1\)-dimensional local domain whose integral closure is a finitely generated \(D\)-module and for non-unit \(a\in D\) denote by \(Z_n(a)\) the set of factorizations of \(a\) into irreducibles with length \(n\). The paper is devoted to an investigation of the structure of the sets \(Z_n(a)\). It is shown in particular that if integers \(k<l\) satisfy \(Z_k(a)\neq\emptyset\), \(Z_l(a)\neq\emptyset\), and the sets \(Z_j(a)\) are empty for \(k<j<l\), then in general \(Z_k(a)\) and \(Z_l(a)\) have a similar structure.