On the ideals of a Noetherian ring
Author:
J. T. Stafford
Journal:
Trans. Amer. Math. Soc. 289 (1985), 381392
MSC:
Primary 16A33; Secondary 16A08, 16A66
MathSciNet review:
779071
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Abstract: We construct various examples of Noetherian rings with peculiar ideal structure. For example, there exists a Noetherian domain with a minimal, nonzero ideal , such that is a commutative polynomial ring in variables, and a Noetherian domain with a (second layer) clique that is not locally finite. The key step in the construction of these rings is to idealize at a right ideal in a Noetherian domain such that is not Artinian.
 [1]
J. Archer, Derivations on commutative rings and projective modules over skew polynomial rings, Ph.D. Thesis, Leeds Univ., 1980.
 [2]
Kenneth
A. Brown, The structure of modules over polycyclic groups,
Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 2,
257–283. MR
600242 (83j:16018), http://dx.doi.org/10.1017/S0305004100058151
 [3]
, Localisation at cliques in group rings (to appear).
 [4]
Jacques
Dixmier, Enveloping algebras, NorthHolland Publishing Co.,
AmsterdamNew YorkOxford, 1977. NorthHolland Mathematical Library, Vol.
14; Translated from the French. MR 0498740
(58 #16803b)
 [5]
A. V. Jategaonkar, Noetherian bimodules, Noetherian Rings and Rings with Polynomial Identity (Proc. Durham Conf.), mimeographed notes, Leeds Univ., 1979.
 [6]
, Localisation in Noetherian rings, London Math. Soc. Lecture Note Series (to appear).
 [7]
Donald
S. Passman, The algebraic structure of group rings, Pure and
Applied Mathematics, WileyInterscience [John Wiley & Sons], New
YorkLondonSydney, 1977. MR 470211
(81d:16001)
 [8]
J.
C. Robson, Idealizers and hereditary Noetherian prime rings,
J. Algebra 22 (1972), 45–81. MR 0299639
(45 #8687)
 [9]
J.
E. Roseblade and P.
F. Smith, A note on the ArtinRees property of certain polycyclic
group algebras, Bull. London Math. Soc. 11 (1979),
no. 2, 184–185. MR 541973
(80h:16008), http://dx.doi.org/10.1112/blms/11.2.184
 [10]
S.
P. Smith, The primitive factor rings of the enveloping algebra of
𝑠𝑙(2,𝐶), J. London Math. Soc. (2)
24 (1981), no. 1, 97–108. MR 623674
(82i:17016), http://dx.doi.org/10.1112/jlms/s224.1.97
 [11]
J. T. Stafford, Generating modules efficiently: Algebraic theory for noncommutative Noetherian rings, J. Algebra 69 (1981), 312346; Corrigendum, ibid. 82 (1983), 294296.
 [12]
, The Goldie rank of a module (to appear).
 [13]
R.
B. Warfield Jr., The number of generators of a module over a fully
bounded ring, J. Algebra 66 (1980), no. 2,
425–447. MR
593603 (82h:16021), http://dx.doi.org/10.1016/00218693(80)900964
 [14]
, Noncommutative, localized rings (to appear).
 [1]
 J. Archer, Derivations on commutative rings and projective modules over skew polynomial rings, Ph.D. Thesis, Leeds Univ., 1980.
 [2]
 K. A. Brown, The structure of modules over polycyclic groups, Math. Proc. Cambridge Philos. Soc. 89 (1981), 257283. MR 600242 (83j:16018)
 [3]
 , Localisation at cliques in group rings (to appear).
 [4]
 J. Dixmier, Enveloping algebras, NorthHolland, Amsterdam, 1977. MR 0498740 (58:16803b)
 [5]
 A. V. Jategaonkar, Noetherian bimodules, Noetherian Rings and Rings with Polynomial Identity (Proc. Durham Conf.), mimeographed notes, Leeds Univ., 1979.
 [6]
 , Localisation in Noetherian rings, London Math. Soc. Lecture Note Series (to appear).
 [7]
 D. S. Passman, The algebraic structure of group rings, Wiley, New York, 1977. MR 470211 (81d:16001)
 [8]
 J. C. Robson, Idealizers and hereditary Noetherian prime rings, J. Algebra 22 (1972), 4581. MR 0299639 (45:8687)
 [9]
 J. E. Roseblade and P. F. Smith, A note on the ArtinRees property of certain polycyclic group algebras, Bull. London Math. Soc. 11 (1979), 184185. MR 541973 (80h:16008)
 [10]
 S. P. Smith, The primitive factor rings of the enveloping algebra of , J. London Math. Soc. 24 (1981), 97108. MR 623674 (82i:17016)
 [11]
 J. T. Stafford, Generating modules efficiently: Algebraic theory for noncommutative Noetherian rings, J. Algebra 69 (1981), 312346; Corrigendum, ibid. 82 (1983), 294296.
 [12]
 , The Goldie rank of a module (to appear).
 [13]
 R. B. Warfield, Jr., The number of generators of a module over a fully bounded ring, J. Algebra 66 (1980), 425447. MR 593603 (82h:16021)
 [14]
 , Noncommutative, localized rings (to appear).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507790715
PII:
S 00029947(1985)07790715
Keywords:
Noetherian rings,
ideal structure,
idealizers,
links between prime ideals and localisation
Article copyright:
© Copyright 1985
American Mathematical Society
