Unique factorization in modules and symmetric algebras

Abstract:

Necessary and sufficient conditions are given for a torsion-free module M over a UFD D to admit a smallest factorial module containing it. This factorial hull is $\cap {M_P}$, the intersection taken over all height one primes of D. In case M is finitely generated, the hull is ${M^{ \ast \ast }}$, the bidual of M. It is shown that if the symmetric algebra ${S_D}(M)$ admits a hull, then the hull is the smallest graded UFD containing ${S_D}(M)$. ${S_D}(M)$ is a UFD if and only if it is a factorial D-module. If M is finitely generated over D, but not necessarily torsion-free, then ${ \oplus _{i \geqslant 0}}{({S^i}(M))^{ \ast \ast }}$ is a graded UFD. Examples are given to show that any finite number of symmetric powers of M may be factorial without ${S_D}(M)$ being factorial.