Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A change of rings theorem and the Artin-Rees property


Author: M. Boratyński
Journal: Proc. Amer. Math. Soc. 53 (1975), 307-310
MSC: Primary 16A60
DOI: https://doi.org/10.1090/S0002-9939-1975-0401840-0
MathSciNet review: 0401840
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A two-sided ideal $ \mathfrak{A}$ of the ring $ R$ is said to have the left $ AR$ property if for every left ideal $ I$ and every $ k$ there exists an $ n$ such that $ {\mathfrak{A}^n} \cap I \subset {\mathfrak{A}^k}I$. Let $ R$ be a left noetherian ring and $ \mathfrak{A}$ a two-sided ideal contained in its Jacobson radical. If $ \mathfrak{A}$ has the $ \operatorname{AR} $ property then $ {\text{l}}\;{\text{gl}}\;{\text{dim}}\;R \leqslant {\text{p}}\;{\text{dim}}\;R/\mathfrak{A} + {\text{l}}\;{\text{gl dim }}R/\mathfrak{A}$, where $ {\text{p}}\;{\text{dim}}\;R/\mathfrak{A}$ denotes the (left) projective dimension of the module $ R/\mathfrak{A}$.


References [Enhancements On Off] (What's this?)

  • [1] N. Bourbaki, Éléments de mathématique. XXIII. Part 1. Les structures fondamentales de l'analyse. Livre II: Algèbre. Chap. 8, Actualités Sci. Indust., no. 1261, Hermann, Paris, 1958. MR 20 #4576. MR 0098114 (20:4576)
  • [2] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
  • [3] J. C. McConnell, Localisation in enveloping rings, J. London Math. Soc. 43 (1968), 421-428. MR 37 #4112. MR 0228532 (37:4112)
  • [4] K. L. Fields, On the global dimension of residue rings, Pacific J. Math. 32 (1970), 345-349. MR 42 #6049. MR 0271166 (42:6049)
  • [5] A. V. Jategaonkar, Injective modules and localization in non-commutative noetherian rings, Trans. Amer. Math. Soc. 190 (1974), 109-123. MR 0349727 (50:2220)
  • [6] Y. Nouazé and P. Gabriel, Idéaux premiers de l'algèbre enveloppante d'une algèbre de Lie nilpotente, J. Algebra 6 (1967), 77-99. MR 34 #5889. MR 0206064 (34:5889)
  • [7] L. W. Small, A change of rings theorem, Proc. Amer. Math. Soc. 19 (1968), 662-666. MR 36 #6460. MR 0223412 (36:6460)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 16A60

Retrieve articles in all journals with MSC: 16A60


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0401840-0
Keywords: $ \operatorname{AR} $ property, projective dimension, Jacobson radical
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society