Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-07-05T09:16:43.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Polynomial Modules Over Macaulay Modules

Published online by Cambridge University Press:  20 November 2018

Robert Gordon
Robert Gordon*
Affiliation:
Department of Mathematics, Temple University, Philadelphia, PA 19122, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [2] we introduced a concept of a Macaulay module over a right noetherian ring by saying that all associated primes of the module have the same codimension. That is to say that a module M over a right noetherian ring R is Macaulay if K dim R/P = K dim R/Q for all P, Q ∈ Ass M. Our main aim here is to extend Nagata’s useful result [6], that Macaulay rings are stable under polynomial adjunction, to a noncommutative setting. Specifically, we prove where x is a commuting indeterminate, that the polynomial module M[x] = M⊗RR[x] is a Macaulay R[x]-module if and only if M is a Macaulay R-module. But actually, we prove a more general result. We show that when M is any module over a right noetherian ring, the associated primes of M[x] are precisely the extensions of the associated primes of M.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 2 , 01 June 1976 , pp. 173 - 176
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Gordon, R., Artinian quotient rings of FBN rings, J. Algebra 35 (1975), 304–307.10.1016/0021-8693(75)90052-6CrossRefGoogle Scholar
2. Gordon, R., Primary decomposition in right noetherian rings, Communications in Algebra 2 (1974), 491–524.10.1080/00927877408822023CrossRefGoogle Scholar
3. Gordon, R. and Robson, J. C., Krull dimension, Memoirs Amer. Math. Soc. 133 (1973).Google Scholar
4. Jategaonkar, A. V., Principal ideal theorem for noetherian P.L rings, J. Algebra 35 (1975), 17–22.10.1016/0021-8693(75)90032-0CrossRefGoogle Scholar
5. Matlis, E., 1-Dimensional Cohen-Macaulay Rings, Lecture Notes in Math. 327, Springer-Verlag, Berlin/New York, 1973.10.1007/BFb0061666CrossRefGoogle Scholar
6. Nagata, M., The theory of multiplicity in general local rings, Proc. of the International Symposium on Algebraic Number Theory, Tokyo-Nikko, 1955, Scientific Council of Japan, Tokyo (1956), 191–226.Google Scholar
7. Nagata, M., Local Rings, Interscience Tracts in Pure and Applied Math. 13, J. Wiley and Sons, New York/London, 1962.Google Scholar