Let R be a ring (with identity). We shall call R a local ring if R is aright noetherian ring such that the Jacobson radical M is a maximal ideal (and so is the only maximal ideal), and R/M is a simple artinian ring. A local ring R with maximal ideal M is called regular if there exists a chainof ideals Mi of such that Mi–1/Mi is generated by a central regular element of R/Mi (1 ≦ i ≦ n). For such a ring R, Walker [6, Theorem 2. 7] proved that R is prime and n is the right global dimension of R, the Krull dimension of R, the homological dimension of theR-module R/M and the supremum of the lengths of chains of prime ideals of R. Such regular local rings will be called n-dimensional. The aim of this note is to give examples of regular local rings. These arise as localizations of universal enveloping algebras of nilpotent Lie algebras over fields and localizations of group algebras of certain finitely generated finite-by-nilpotent groups.
@article{10_1017_S0017089500002792,
author = {Smith, P. F.},
title = {On non-commutative regular local rings},
journal = {Glasgow mathematical journal},
pages = {98--102},
year = {1976},
volume = {17},
number = {2},
doi = {10.1017/S0017089500002792},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500002792/}
}
TY - JOUR
AU - Smith, P. F.
TI - On non-commutative regular local rings
JO - Glasgow mathematical journal
PY - 1976
SP - 98
EP - 102
VL - 17
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500002792/
DO - 10.1017/S0017089500002792
ID - 10_1017_S0017089500002792
ER -
%0 Journal Article
%A Smith, P. F.
%T On non-commutative regular local rings
%J Glasgow mathematical journal
%D 1976
%P 98-102
%V 17
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500002792/
%R 10.1017/S0017089500002792
%F 10_1017_S0017089500002792
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