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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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.
References
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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