A surprising characterization of unique factorization domains

Abstract: A surprising characterization of unique factorization domains \\ Abstract: & We address some recent work on the generalization of the UFD propery which has pointed back to an open problem first mentioned in a paper by myself, Dan Anderson, Muhammad Zafrullah, and Franz Halter-Koch (Criteria for unique factorization in integral domains, J. Pure Appl. Algebra 127(1998), 205--218), which we abbreviate as ACHKZ. Fix a positive integer $n>1$. Call an atomic integral domain $D$ quasi-$n$-factorial if, for any irreducible elements $x_1, \ldots , x_n, y_1, \ldots , y_n$, the equality $x_1\cdots x_n=y_1\cdots y_n$ implies that $x_i=u_iy_{\sigma(i)}$ for some unit $u_i$ and permutation $\sigma$ of $\{1,\ldots ,n\}$. Further, $D$ is length-factorial if it is quasi-$n$-factorial for all $n>1$. Jim Coykendall and William W. Smith showed in 2011 the surprising result that an atomic monoid is a UFD if and only if it is length-factorial. This allows one to alter the classic definition of a UFD. in a surprising manner. The authors in ACHKZ offer examples of monoids which are quasi-$n$-factorial for specific $n$, but are not factorial. They offer no such example of an integral domain. Hence, the Coykendall-Smith result makes the following problem explored in ACHKZ all the more relevant. Open Problem: Does there exist an atomic integral domain $D$ which is quasi-$n$-factorial for some $n>1$, but not factorial?

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)