Variations on Unique Factorization Domains

In the papers Factorization in integral domains (J. Pure Appl. Algebra 69 (1990), 1–19) and Factorization in integral domains, II (J. Algebra 152 (1992), 78–93), Dan Anderson and his co-authors David Anderson and Muhammad Zafrullah, established much of the foundation for studying the factorization properties of rings and integral domains that remains active in the literature to this day. Hence, I will take this opportunity to talk about some recent work on the generalization of the UFD property which has pointed back to an open problem first mentioned in a paper by myself, Dan, Muhammad, 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 x₁, … , x_n, y₁, … , y_n the equality

implies that x_i = u_i y_σ(i) for some unit u_i and permutation σ of {1, … , 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. The authors in ACHKZ offer examples of monoids which are quasi-n-factorial for specific n, but 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?