@Article{         10.2307/20161816,
  Author        = "Aschenbrenner, Matthias and Hillar, Christopher J.",
  Abstract      = {Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let $\germ{S}\_{X}$ be the group of permutations of X. The group $\germ{S}\_{X}$ acts on R in a natural way, and this in turn gives R the structure of a left module over the group ring $R[\germ{S}\_{X}]$ . We prove that all ideals of R invariant under the action of $\germ{S}\_{X}$ are finitely generated as $R[\germ{S}\_{X}]$ -modules. The proof involves introducing a certain well-quasi-ordering on monomials and developing a theory of Gr{\"o}bner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finite-dimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it.},
  date-added    = "2022-09-26 22:22:57 +0200",
  date-modified = "2022-09-26 22:22:57 +0200",
  ISSN          = "00029947",
  Journal       = "Transactions of the American Mathematical Society",
  Number        = "11",
  Pages         = "5171--5192",
  Publisher     = "American Mathematical Society",
  Title         = "Finite Generation of Symmetric Ideals",
  URL           = "http://www.jstor.org/stable/20161816",
  URLDate       = "2022-09-26",
  Volume        = "359",
  Year          = "2007",
  bdsk-url-1    = "http://www.jstor.org/stable/20161816",
  File          = "FINITE GENERATION OF SYMMETRIC IDEALS - S0002-9947-07-04116-5 - a.pdf",
  file-2        = "FINITE GENERATION OF SYMMETRIC IDEALS - a.pdf"
}

@Article{ 10.2307/20161816, Author = "Aschenbrenner, Matthias and Hillar, Christopher J.", Abstract = {Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let $\germ{S}_{X}$ be the group of permutations of X. The group $\germ{S}_{X}$ acts on R in a natural way, and this in turn gives R the structure of a left module over the group ring $R[\germ{S}_{X}]$ . We prove that all ideals of R invariant under the action of $\germ{S}_{X}$ are finitely generated as $R[\germ{S}_{X}]$ -modules. The proof involves introducing a certain well-quasi-ordering on monomials and developing a theory of Gr{\"o}bner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finite-dimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it.}, date-added = "2022-09-26 22:22:57 +0200", date-modified = "2022-09-26 22:22:57 +0200", ISSN = "00029947", Journal = "Transactions of the American Mathematical Society", Number = "11", Pages = "5171--5192", Publisher = "American Mathematical Society", Title = "Finite Generation of Symmetric Ideals", URL = "http://www.jstor.org/stable/20161816", URLDate = "2022-09-26", Volume = "359", Year = "2007", bdsk-url-1 = "http://www.jstor.org/stable/20161816", File = "FINITE GENERATION OF SYMMETRIC IDEALS - S0002-9947-07-04116-5 - a.pdf", file-2 = "FINITE GENERATION OF SYMMETRIC IDEALS - a.pdf" }