Skip to Main Content

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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Unique factorization in modules and symmetric algebras
HTML articles powered by AMS MathViewer

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


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.
  • Nicolas Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR 0360549
  • D. L. Costa, Symmetric algebras and retracts, Dissertation, Univ. of Kansas, 1974.
  • Robert M. Fossum, The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74, Springer-Verlag, New York-Heidelberg, 1973. MR 0382254
  • László Fuchs, Infinite abelian groups. Vol. II, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973. MR 0349869
  • Daniel Lazard, Autour de la platitude, Bull. Soc. Math. France 97 (1969), 81–128 (French). MR 254100
  • Anne-Marie Nicolas, Modules factoriels, Bull. Sci. Math. (2) 95 (1971), 33–52 (French). MR 284426
  • Anne-Marie Nicolas, Extensions factorielles et modules factorables, Bull. Sci. Math. (2) 98 (1974), no. 2, 117–143 (French). MR 424788
  • Pierre Samuel, Anneaux gradués factoriels et modules réflexifs, Bull. Soc. Math. France 92 (1964), 237–249 (French). MR 186702
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 13F15
  • Retrieve articles in all journals with MSC: 13F15
Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 224 (1976), 267-280
  • MSC: Primary 13F15
  • DOI:
  • MathSciNet review: 0422250