## Unique factorization in modules and symmetric algebras

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- by Douglas L. Costa PDF
- Trans. Amer. Math. Soc.
**224**(1976), 267-280 Request permission

## 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.

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## Additional Information

- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**224**(1976), 267-280 - MSC: Primary 13F15
- DOI: https://doi.org/10.1090/S0002-9947-1976-0422250-1
- MathSciNet review: 0422250