Unique factorization in modules and symmetric algebras

Douglas L. Costa
1976 Transactions of the American Mathematical Society  
Necessary and sufficient conditions are given for a torsionfree module M over a UFD D to admit a smallest factorial module containing it. This factorial hull is C\Mp, the intersection taken over all height one primes of D. In case M is finitely generated, the hull is M**, the bidual of M. It is shown that if the symmetric algebra Sp(M) admits a hull, then the hull is the smallest graded UFD containing SD(M). SD(M) is a UFD if and only if it is a factorial D-module. If M is finitely generated
more » ... nitely generated over D, but not necessarily torsion-free, then ©jxjistyw))** is a graded UFD. Examples are given to show that any finite number of symmetric powers of M may be factorial without Sp(M) being factorial.
doi:10.1090/s0002-9947-1976-0422250-1 fatcat:omv5uy5fbveozc7jt2ojwujh6q