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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Unique factorization in modules and symmetric algebras

Author: Douglas L. Costa
Journal: Trans. Amer. Math. Soc. 224 (1976), 267-280
MSC: Primary 13F15
MathSciNet review: 0422250
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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

PII: S 0002-9947(1976)0422250-1
Keywords: Factorial module, symmetric algebra
Article copyright: © Copyright 1976 American Mathematical Society

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