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 ∩ M P \cap {M_P} , the intersection taken over all height one primes of D. In case M is finitely generated, the hull is M ∗ ∗ {M^{ \ast \ast }} , the bidual of M. It is shown that if the symmetric algebra S D ( M ) {S_D}(M) admits a hull, then the hull is the smallest graded UFD containing S D ( M ) {S_D}(M) . 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 ⊕ i ⩾ 0 ( S i ( M ) ) ∗ ∗ { \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 ) {S_D}(M) being factorial.